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.
I would suggest that what I consider to be "outside the cave" might simply be a larger cave, but yes, there is an urge to show people the new space.
I don't think anything comes after death. After death, there is no longer a person who might experience anything.
It's difficult for most people, including me, to properly wrap their heads around the idea of not existing. In the absence of an afterlife, many people still imagine themselves persisting forever, alone in darkness and silence. It's an idea I find more frightening than Hell. (Therefore if there is a Hell, that's how it might well manifest for me.) I remind myself that at the moment of death, there won't be a me anymore, so any thoughts of what might happen to me after death are meaningless.
The energy in the body is eternal, but it's not a soul. It's the simple chemical and electrical energy in our bodily fluids, as well as our muscles and our brains. When we die, it escapes mostly as heat. It also contributes to the reactions which decompose the body. Some of it goes to feed some lucky earthworms, and eventually to enrich the soil around the grave and whatever grows in it.
The Christian concept of a soul has nothing to do with quantifiable energy. (They did try to connect the two by identifying a measurable loss of substance at the moment of death, but ultimately failed.) The soul, hypothetically, is an intangible entity which accompanies the human body for its entire lifespan. It controls the body and supplies it with human qualities, and human worth. Essentially the soul is an externalisation of certain brain functions and social conventions. Upon death, it severs all connections with the physical world and goes to God to be distributed as He sees fit.
The soul shares a category with Buddhism's karma and all manner of energies posited by New Age systems, in that although it's credited with many effects (for example a person's conscience), none can be attributed to it unambiguously. Associating it with the ordinary, observable energy inside the body is one way to make it sound more plausible, but there's no merit in the comparison.
It is sad to think that many people who live selfishly will never be punished for it in their lifetimes, and then receive no punishment after death. That's why it's important to work to bring them to account while they're alive. The idea that it would be right, or just, for them to be judged when they die will not make it happen. It can be nicer to imagine, but that doesn't make it true. It's up to living people to uphold justice as we see it. Take it as motivation, not discouragement.
Generally our personal beliefs and values are passed on to us by our parents and friends, and reinforced by scoiety. Fair or not, that's how it is. You always have the option to go against those beliefs and values, but then you're going against your family, your friends, your society and often your own instincts. It is possible, and on rare occasions it's the best thing to do. It's what most religious people face when they want to leave their religions, for example.
There's no afterlife in an atheist's worldview. That's why I value life so much. Since there's no reward waiting for me, I work to have my rewards while I'm still here. That needn't mean living selfishly, as charity and altruism can be their own reward. If I believed in a heavenly afterlife I could reach, I might be satisfied with a less wonderful life because I'd think something better was coming. Again, the fact that it's a nice idea does not make it true. So I try to make my heaven right here.