It is most certainly not your problem, but can you really let him slip down a path that might be very difficult to recover from he was told he couldn’t stay, was furious, and then got on a train and turned up at the OP’s house at a time where she felt she couldn’t refuse to let him stay.
Nobody knows anything about him other than what he’s told the OP’s daughter.
He claims he was in an abusive situation but his actions point more to his being the abusive, manipulative one than the other way around. And before someone shouts “he’s a child,” 16 year olds are perfectly capable of being abusive and manipulative.
Added to which, he apparently has no-one. No friends, no family, no workmates, nothing? Why is that? And if he has no-one, where has he been all day then? 
And tbh if it was my daughter I’d be reading her the riot act as well and telling her that I’d said no, that I was the adult, and that if she had cooked up this plan with her bf for him to stay then she was way out of line and there would be consequences. I would also tell her that if he’d pressured her into this she should tell me.Either way he’d be out on his arse tonight, no questions asked.
Where he goes is not the OP’s problem. He could ring SS but he doesn’t want to. Well tough shit. It’s a fact of life that sometimes you have to rely on the authorities, and the OP isn’t responsible for him.