OP, nobody kicks a 14 year old child out of the house just because their sibling wants a room to themselves. If, and it's a big if, the child is genuinely being kicked out, there is clearly a lot more going on here. You should think long and hard, and get professional advice, before getting in the middle of it.
For starters, the boys parents would be breaking the law.
Pump the brakes though...Have you actually spoken to the parents yet? It sounds like the boy doesn't have a great home life - This could simply be a case of bad parenting in that they've told him they're chucking him out as a threat (without meaning it) and he's taken it literally.
Also you say they're not great parents and you often feed/look after the kid...could it be that the boy and your daughter have told his parents that you've offered to take him in?
Bless you for wanting to look after the child. Having him stay with you wouldn't be illegal, but you must involve social services. But you also need to consider the effect this could have on your child(ren).
Moving in with a partner full time is a big, intense situation for most adults. For kids, especially at their age and with whatever the boy has going on...there's potential for it to do lasting damage.