I think in the first place, it would be best for this topic to not come up in casual conversation. If you and your friends are college students or recently graduated I imagine a frequent topic of conversation is, "What's your major? Where did you go to college?" And if someone says they didn't
go to college, they're expected to say what they did instead, like, "I started working at XYZ right after high school," "I joined the Army," "I did some traveling with a non-profit organization," etc.. At least, this has been my experience.
It's great that your BF wants to be honest about where he was during the typical college years, but as you have seen this goes over like a lead balloon in a casual conversation and abruptly derails it. I would consider coming up with something vague to say instead, possibly leading into another topic such as his plans for the future. For example, if he could say, "I didn't have the chance to go to college right out of high school, which I regret. However, I'm now attending Community College part-time working on my ABC degree." (Obviously he should only say this if it were true.)
If it's a less casual conversation, like with a good friend of yours, I think emphasizing the positive and the future is still the way to go, even if you add more details about the past. "He had XYZ issues [if BF is okay with you sharing this] when he was younger, and he did some stupid things because of it. But he's really grown up so much since then, and done these specific things to help him overcome his issues. Now he's attending Community College part-time to get his degree while working at the ABC business."
Personally, I feel like this is the sort of issue where, if you mention it at all, you need to discuss it further; in other words, beandip doesn't work well, because abruptly changing the subject makes people think you're hiding something or that something is really wrong. (I think beandip would be appropriate if you were, say, talking to Person A and Person B brought up the felon issue, which Person A didn't know about and which you didn't plan to talk about with them.) So don't bring it up at all to those who don't know and don't need to know; and with closer people, be prepared to admit the wrongdoing but emphasize the steps that were taken to improve the situation, and the positive plans for the future. That's my advice at least, I hope it's helpful to you.