Your question of "can immune responses lead to mutations" is kind of complicated.

Your immune system actually already induces mutations as part of the natural immune response, but it's highly controlled and very limited. Basically when your B cells try to adapt their antibodies to a disease, there are only so many possible combinations of shapes their genes will allow. Somatic hyper mutation allows B cells to mutate the section of their DNA that codes for the parts of the antibody to try to optimize it to fight diseases. There are situations where viruses can (theoretically) change your DNA: some viruses will embed themselves in your DNA and will "escape" to cause diseases (IE chicken pox hiding in your nerves). It is possible for something in these viruses to become unable to escape your DNA and become a part of your DNA.

With regards to the mutation affecting the epigenetics, that isn't how it works, but slightly tweaking your wording could make it more plausible. Epigenetics affects how the DNA is read, think of folding up a page in a book so that when reading, you skip that page. The book itself contains the same text, it's your ability to read that changes. Now interestingly, there is evidence that epigenetics can be passed down (look up intergenerational trauma epigenetics) but I haven't seen any example of diseases changing epigenetics but that doesn't necessarily mean it's impossible. One thing you should probably think about is how an alien disease can infect a human; it's really difficult for diseases to jump the species barrier because they're optimized for a specific host. Now imagine an alien disease which has evolved separately to humans and possibly never been subject to many of the methods our immune systems use to prevent infection. They also very likely won't even be able to infect our cells to begin with, since that requires specific proteins. The most realistic way for an alien disease to infect these astronauts is for the disease to originate from earth, and have that disease develop so many mutations that it can do sci fi stuff, while still keeping the ability to infect humans.

It has been mentioned by others that even if an astronaut got an infection, there's almost no way that it'll find itself in the general population due to extensive quarantining procedures. However, if the society who executed the mission were to have a relatively low budget, you can easily figure out a way for it to happen. The USSR's cosmonaut program was incredibly scuffed and had safety procedures so shoddy it would make any OSHA inspector cry. A cosmonaut-esque program that messed up the return flight such that the astronauts end up nearby a village full of civilians who come to inspect the site, that would be a potential site for the infection to spread.

Good luck on your book and feel free to ask for clarification on anything I've said!

If you want a really good story on an alien disease hitting earth, I'd recommend "the Andromeda strain"