If my boyfriend wants to “give” me his old car to trade in when I purchase a new car, would he have any legal rights to the new car?

I agree with the previous lawyer. You'd better off if you reduce your agreement to writing saying that the old car is a gift. 

If his name is not on title, and especially if he's not co-signing the loan, then it wouldn't be his car.  As I understand the facts, he is giving you a gift.  (Beware: there could be possible tax consequences of such a gift; you should consult with a tax advisor on that issue.)  However, judges are just people, and if you asked your question to 100 judges, some of them might rule in your boyfriend's favor.  This is especially true if your boyfriend were to lie in court and say you had an agreement that he would be part owner, or that you'd pay him back, etc.  Anyone can create a problem if they're willing to lie.  In that case, this would really be an "evidentiary" issue.  What I would do to get around that is email him and clarify - even in an informal written exchange - that he is 'giving' you the car.  E.g., before you purchase the car, email him saying, "Just wanted to say how sweet it is that you are giving me your car to use as a trade in.  I know you could have sold it and gotten some money for it, and I really appreciate your generosity."  Or a question asking him if he's sure he wants to 'gift' it to you, etc.  If he doesn't respond to that email and say something like, "What are you talking about?  We had an agreement that you'd pay me" or something, then it really limits his ability to lie about your arrangement in court a couple years from now.  Or else you could simply draft up an agreement of sorts that spells out exactly what you both intend, if you feel comfortable.  Good luck.