I hope so, 'cause I’ve got a list! 
I’ll hang with Vergil until he starts to recite his incredibly dull poetry.
Mind you, I’m just saying that I don’t think Stark should be allowed in the first place. I’m not saying that he would be a good choice if he were allowed.
Hawkeye is an archer type, right? That might be a good idea: It’s probably easier to recover or make arrows than it is bullets, in the post-apocalyptic world.
I’d be more worried about HIM going hungry.
Anyway, how about John Clark from Tom Clancy’s novels? Ex-seal, high or low tech survival skills, seems like good company if you stay on his good side.
I’d choose Zakalwe from Iain Banks’ “Use of Weapons” for survival. Whatever side he is on WILL be the winning side, and if there are no sides, he’ll make up his own. Does NOT matter who the opponent is, he will beat them, because that is who he is and what he does.
For breeding, I’d go with Clara from The Guild. There’s a cute, friendly, sexy blonde with a uterus like a machine gun under all that weight, and post-apocalyptic civilizations are GREAT for dieting, being as how food is hard to find. Of course, somebody ELSE is gonna have to watch the kids.
How’s about Max Klinger? He’s a barbeque expert, would have a thriving barter economy going in a trice, and the mutants would never be quite sure whether he was one of them. And given a couple of tries, he can even pour a cement floor for the hut. . . plus I’m sure he’s picked up a modicum of medical knowledge over the years.
I dunno about Hawkeye. Once his hearing aid’s batteries run out, it’s gonna be a bitch replacing them post-apoc, and that’s gonna limit him severely.
I’m pretty sure he came back from the dead with full hearing. Not that I’ve seen anything to indicate it; it would just be typical.
'Sides, like I said, it’s less that he’s the ideal choice, just that he’s better company than Tony Stark.
Jeremiah Johnson
Then pick Trapper John. 
Didn’t you see the first movie? Max pre-dated the apocalypse. I would pick him him if I wasn’t worried that he would break into a rant against Jews in the middle of a firefight.
So I’ll pick Milla Jovovich as Alice. Smokin’ Hot Death on Wheels. Or Sienna Guillory, her pal in the sequel. If I can get both – to preserve body heat – even better.
Conrad. OK, he may be a god but then again, he might not be. He’s probably just a really long-lived mutant. But not super-powered by any means – just a survivor. And, unlike some other survivor types like Snake Plishken or Hell Tanner, he won’t sell you to the first mutant slave-trader for a bottle of water and a handful of cigars.
Well, if I was going to pick anyone from MASH, it’d probably be Hotlips.
Joe Pike. If feminine companionship were considered, 7 of 9. Not sure Buckaroo is ruthless enough.
Half-Cock Jack Shaftoe. Sure, he’s an illiterate 17th century vagabond and street ruffian with half a penis, but he’s surprisingly resourceful in emergencies, is a highly imaginative strategist, and never, ever hesitates. Plus, most of the people who tag along with him tend to survive, which is more than you can say about most of the people mentioned in this thread.