If your bf is currently a research fellow on a J visa, he could simply try to find a similar research job at your medical school and then transfer / get a new visa. I don't know the exact details, but overall it's not too difficult. Of course, he may like his current position and there is no guarantee that any new position would be better.
As far as getting a residency, he needs to do what all IMG's need to do -- take and do well on the USMLE's, try to get US clinical experience, and then apply in the match. He could do so now, or he could plan to match the same year that you match and you could couples match.
As far as visas for residents, he would need to do so on a J or an H. The J visa for residency is different from his current visa. There is a thread about visas for residency stickied at the top of this forum, and you should review that.
Of course, one way to get around the visa issue altogether is to marry him. That would get him a green card, although you should absolutely talk to an immigration lawyer prior to thinking of getting married because bad things can happen -- you can trigger a "marriage of convenience" investigation and that could be very bad. Hence, talk to a lawyer first to do it right.
