It's a Terrible Life did a WONDERFUL job resolving this, but I had this idea before the episode aired so this is my take on how Dean might have been convinced.
"Well, I guess I'm not the man either of our dad's wanted me to be. Find someone else. It's not me," Dean said just before he let blissful unconsciousness take him over.
Castiel watched Dean and hated what he had done to him. He had driven Dean to this. Anna had been right. He had been too late to question the orders that had brought Dean to this. Sam had been right. It had all been a waste. Castiel might have just sealed their fates by destroying the psyche of the only person who could save them all.
"I'm sorry, Dean," Castiel said softly. "I did this to you and I don't know how to undo it."
"I do," said a soft feminine voice behind him.
Castiel turned and saw Anna there.
"You were right all along. I'm sorry."
"Don't be. You forget. I went through what you are feeling now. The doubt and the pain that comes with it, it's real, but necessary."
"Doesn't matter any more. I don't matter anymore. Even if Uriel was the instigator, I still disobeyed."
"No, you fought back, you stayed true to God even with your doubts. You confronted Uriel with that certainty and defied him, not God," Anna comforted.
"But not soon enough to save Dean. I've committed the worst act possible. I allowed an atrocity to happen to a righteous man, a good man for no good reason. I didn't save Dean and now, Dean will abandon us. Who could blame him? The world will end because I wasn't strong enough."
Anna approached Castiel and cupped his cheek.
"You're beginning to sound like him, you know. I see his influence on you already," Anna said, a soft smile on her face. "Dean once challenged me on my decision to fall, stating all the weaknesses and frailties of humanity and yet, if given the opportunity again, I'd take it. No, Castiel, you are strong, so is Dean, perhaps the both of you can find strength in each other."
Castiel looked into her eyes, his own welling.
"How can I help him now?"
"Maybe we both can. I have resources now that I'm back in the fold. I don't fear any repercussions. If they were to happen, they already would have. I have to believe that I was spared punishment because I can be of some use to you and Dean."
"What do you suggest?"
"I know of someone who is willing to help, wants to help."
"Who could that be? I fear that no one, not even Sam could sway Dean now."
"His father could," Anna declared matter of factly.
"I do not understand."
"Dean's father, John Winchester, he knows of his son's suffering and feels he can help him."
"Anna, it was one thing to place Dean in the past so that he could know the truth about his mother sealing their destinies. He could not exact change there. He will question any vision of his father placed before him."
"Perhaps, at first, but John Winchester is the only man who can tell Dean the truth about what happened to him in Hell. Dean derives his existence, of the man he is to his father's influence. Once convinced of his reality, John is the only voice Dean will listen to."
"I hope that Dean will believe," Castiel said.
"Have faith that he will, Castiel," Anna replied as she placed her finger on Dean's forehead.
Dean woke up feeling groggy and in pain. He looked over to where he last remembered seeing Castiel and the chair was empty. He looked at the end of the bed to find Sam fast asleep, staying by Dean's side. After everything they had gone through, Dean still found himself glad to see Sam there. He couldn't remember what had happened after he had passed out, but he hoped Sam would be honest with him about what had happened to Alistair.
A cold shiver came over him and it brought the hairs on his neck up. He turned towards the source and went white with shock and horror.
"Dad?" Dean said in hushed surprise. "I must be dreaming."
"You're not dreaming, son," John replied back, a look of longing on his face.
"No! Don't call me son!!! I'm NOT YOUR SON!!! You're NOT real. You're not! CAS!!! I won't let you mess with my head like this!!!" Dean yelled in a mix of anger and despair.
"Dean, please, this is real. I'm really here."
"No, even if this was real, my father would NOT be here."
"Because….because…he'd know what a loser of a son he has…" Dean said, his voice quivering from need and confusion. "That I never measured up."
"That's not true. What Alistair told you was a lie, Dean."
"What if you're the lie, huh? Sent here to make me see that what I did wasn't wrong, that giving into Alistair's offer wasn't weakness, that torturing all those souls wasn't really my fault, that I really didn't like doing it? Well, I don't buy it. You hear me, Cas? I'm not buying this so you can just wake me up from whatever you want to call this because I'm still out. DO YOU HEAR ME? Find someone else!!!"
"There is no one else, Dean," John answered.
Dean looked over at his father, wanting desperately to believe what he was seeing, but his hunter's instincts telling him to question it. He hadn't aged since he had died.
"Please don't do this. Haven't I done enough? Haven't I screwed up enough? I can't freakin' save the world, okay? I don't have it in me."
"Yes, Dean, you do. More than you're giving yourself credit for. I blame myself for that."
"Just go away, okay?" Dean pleaded.
Dean watched Sam and he hadn't moved. He must be in a dream if Sam hadn't awakened by now.
"No, I won't, Dean. You need to listen to me," John said, but without any of the command timbre in his voice.
"Dad, if you really are my dad, you can pep talk me all you want, but nothing you can say can change things. It's my fault that we're in the mess we're in. I caused this. Those are the facts. Nothing can change them!"
John listened to his son, tormented by everything that had happened and hated that in the end, Dean would have to shoulder saving the world all on his own, just like he had shouldered everything else in his young life.
"You can change it, Dean."
"Yeh, by saving the world. I'm the ONLY one who can because I'M the one who started it in the first place," Dean scoffed.
"Breaking isn't a weakness. It's inevitable in Hell."
"You didn't break, Dad," Dean said, his voice cracking with want.
"Yes, I did, son. Alistair lied to you."
Dean looked at his father and his expression turned into a scowl.
"You're a liar. My dad, he was a hero. He was strong. HE would never break. He knew lives were at stake. He would have never let Alistair crack him."
"But I didn't know, Dean. Just like you didn't. And even if I had known, I couldn't have held out, but I know that if you had known what was at stake, you would have. "
Dean started to laugh.
"Now, I know you're not my father. He knows me better than that."
"You're right, I do know you better. It's you who doesn't realize how strong a man you are, how good a man you are. You never did the hunts for revenge like I did," John paused. "Like Sam does."
"Leave Sam out of this!" Dean protested.
"You did the hunts because you wanted to save people. Save people because you felt they deserved saving, so you could spare them what we went through, but also to save them because you couldn't save your mother."
"I couldn't save her, Dean. I lived with that guilt all my life and without really knowing it, I passed that guilt onto you. You were only four years old. There was nothing you could have done, but still, you felt you could have saved her. Just like you wanted to save her in Castiel's vision of the past. You've been trying to save her all your life, but it wasn't your fault."
Dean felt the tears of that loss hit him and the pain of the memory of losing his mother twice.
"Please stop," Dean pleaded.
"I can't because this is a long time in coming, son. When I talked to you at the hospital, all I had time for was to tell you how proud I was of you. Everything I said then, that was the truth, but I also laid the worst burden a father ever could on your shoulders. Save Sam or kill him."
"Leave Sam out of this!" Dean repeated.
"I can't, Dean because he is a part of this, a part of this family, a part of the man you came to be. What happened to you happened because you are a true righteous man and I can't let you believe that being that, being human who has limits, makes you at fault in all this. Destiny wrote this seal, Dean. It was written long before I was born, you were born and before our family was destroyed by Azazel. Your mother because she loved me so much, because she wanted to have a family, you boys, put into place our destiny when she made her deal and when Azazel came to collect, she fought to protect her family and gave her life to do it. Nothing could have saved her. Her destiny was set in stone years before."
John found himself tearing up in recalling his beloved Mary. It had been a long time since he could think of her without vengeance interfering in the memories.
"I don't believe in destiny! I don't believe that I was destined for anything except to rot in Hell for what I did."
"Well, maybe not, call it what you want, and I would give my soul if I could do it, but Dean, destiny chose you because of the man you are. You were NEVER supposed to end up in Hell."
"Neither were you," Dean said softly.
"That's where you're wrong. I was not a good man. I killed in Vietnam and I don't just mean enemy soldiers. I mean innocent women and children too," John said, his heart filled with regret.
"It was war, Dad," Dean tried to comfort.
"Doesn't make it right, son."
"But I've killed."
"Monsters, demons, yes, but not humans, not innocent humans. Yes, there were possessed humans, but you were releasing them from a life no human should live."
"I don't know, Dad…maybe after what I did in Hell, I do deserve to end up there."
"No, Dean, you don't. You did your best there. It's all anyone can be asked of there. The fact that you feel remorse for what you did there means something. It means something to the angels, it means something to God, and it means something to me. The only thing that drove you to Hell was love, love for your brother and how much you value him and his life. You shouldn't have been punished for that, but the demons…they seized on –"
"That weakness?" Dean said. "Yeh, exactly."
"We all have weaknesses. Without them, we wouldn't be human or would know what to overcome. You're not weak. You just love too much, care too much. Those aren't weaknesses. They are your strengths. Your only weakness is that you give so much you leave nothing for yourself."
Dean ignored his father's compliments, unwilling to acknowledge that any weakness was acceptable. In the end, he had been weak and that was enough for him to know that he couldn't do what they were asking of him to do.
"Dean, I know you're afraid, but you are not weak. You are strong. Strong because you have a conviction I never possessed. You believe in what you do. The family motto you have been living by, that didn't come from me. That came from you. It's what you took from what I taught you. I did the job, but saving people, that was an afterthought. I got just as much, if not more satisfaction from killing evil that I did at saving people. I never told you boys this, but all that hunting I did was a means to an end, to find your mother's killer. I trained you both so you could protect yourselves, but Dean, you gave the work its meaning. Yes, you've doubted whether what you have been doing makes a difference, whether you have made a difference, but believe me, you have, not just from defeating evil, but from the compassion you give to those you have saved. Your love for Sam, for humanity, it flows through you. I'm so –"
"No, don't say it. Don't tell me you're proud of me because the last time you did that you died and then I failed Sam. I let him die, Dad. I didn't save him like you told me to. Then I go and make a deal, go to Hell, push Sam into using his powers by not being there to stop it and what I did in Hell, please Dad, you can't possibly be proud of that. What I did there was unimaginable!"
"But I am proud of you. You can shut out the words, but I know you feel that pride in me for you. You did what you could under terrible circumstances and you chose to sacrifice yourself for Sam, your life and your soul. Sam's choice to use his powers was his own. Nothing you could have done could have stopped him. Sam's destiny runs along a different path, but make no mistake, without you there, regardless of his words or his motives, he will be lost. If you give in, so will he."
"Dad, if you know all this, then tell me how I can stop all this, to stop Sam from going any farther."
"I don't know the future, son. I just know my sons. Sam is too much like me. He's doing what I would have done if I had been chosen as he had."
"No, you wouldn't have –"
"Dean, stop putting me on a pedestal. You know better. You may believe in me, a belief that I don't deserve, but you shouldn't measure yourself by me. I'm not half the man you are, son. I don't even come close and that's why I know you can do this. You are the only man who can do this and more importantly, you are the right man. You need to find that strength from within yourself. Not from me, not from Sam, not from anyone. And you can't let anything, not any demon, not Ruby, not Lillith, not Alistair break that strength because it cannot be destroyed. Look at me," John said.
Dean looked up at his father, his words finally and slowly sinking in. Dean allowed himself to believe that this was his father because he needed to believe it, needed to let his father's words soothe his aching soul.
"I haven't earned the right to make you do anything anymore and the time for orders is over. I can't and I won't make you do anything, but I know that if you listen to your heart, to the soul that survived Hell, you'll know what the right thing to do is, you always have and even though I would give anything if you didn't have to do this, I am just as sure that you will not fail to bring everything you have, everything you are to make it happen. Believe that instead of what the demons tell you, instead of what I have told you in the past, instead of what your hero worship of me tells you. I'm the weak one, Dean, not you."
John placed a hand on Dean's chest.
"Sleep now and believe that I'm with you, here, in your heart. Both your mother and I are. We will never abandon you."
Dean's eyes filled with tears and he let them fall as weariness slowly weighted his eyelids and he succumbed to sleep.
Castiel appeared beside John and watched with relief at the peace on Dean's face.
"Protect him, Castiel."
"I will with my life," Castiel said with conviction.
"And Sam?" John asked hesitantly.
"His fate is still unclear."
"Dean will save him," John said with his own conviction. "If there's a way, Dean will find it."
"I hope so."
John then disappeared. Castiel continued to watch Dean sleep and couldn't help, but feel hopeful again. He had failed Dean once, but it would not happen again, he had resolved. Even if it meant defying orders, he would keep his promise to Dean's father. He would protect his son. To the end, whatever that end might be.
FIN. Hope you enjoyed my take. Thanks for reading and reviews are always welcome!