Infinity/eternity is frightening to anyone who really thinks about it. It can be especially troubling to the recently ex-religious, because they've lost their particular assurances of perfect eternal happiness. Compared to that superficial idea, the prospect of simple unending persistence is much harder to shake, because we all have a tough time imagining ourselves not existing.
Your boyfriend's on the right track intellectually, because he accepts that there probably won't be a him after he dies. We just need to address that small probability that there will, and settle his emotions a bit.
Right then, what if he IS wrong? Anything is possible. The odds of any particular afterlife story being the real deal, including whichever version he may have grown up with, is almost zero. That's because there are an infinite number of possible afterlifes: the thousands imagined by humans in the last few thousand years, plus the countless ones we haven't thought of yet.
You haven't said what he specifically fears about living forever. Maybe he hasn't really told you. He might fear torture in hell, or a post-life sentence as a ghost watching everything he knows crumble away, or imprisonment in his own disembodied mind deprived of all senses, going mad in darkness and silence. That last one sure gives me the willies.
If anything is possible, all of the above are. Is any one of them at all likely? No, because they're swimming in an endless sea of alternatives. Put simply, whatever he's afraid his ultimate fate will be if he doesn't end at death, he's almost certainly wrong.
The only death scenario for which there is any solid physical support is oblivion. The mind, the ego, the identity, the memory and everything else that comprises "you" is contained and operated in a physical, bio-electrical brain. After death, that brain first shuts down and then disintegrates. It's a materialist view, I know, but arguments against it centre on things the physical brain supposedly can't do, and without exhaustive analysis are arguments from ignorance.
So take your boyfriend's most troubling scenario and work out the chances: the probability of an identity outliving the human brain outside of all detection, multiplied by the probability of that scenario being the one real one out of infinity. It's like the chances of finding a particular blade of grass if you don't know which country it's in.
That's the thrust of my recommended reassurance. To sum up, if there's an afterlife, you can be confident that it's different from and therefore less scary than his idea of it.