If I was an intern at one of the MBB firms and didn't receive an offer to return, how would that affect my full-time recruitment for other MBB firms?

So the question is, will they be open to it? Absolutely.

Will it impact your candidacy? Yes.

The first thing you do, is make sure you are being honest with all parties involved. Be prepared to provide a thoughtful answer about why you don’t think you got the offer, what you learned from the experience, and why you think you will be a good consultant.