I am trying to find the title of a movie with an elderly man living in a retirement home telling his life story of how he can't die and he has a pet mouse that he keeps in a shack out back of the retirement home. The mouse also can't die.

That sounds like just the last few minutes of The Long Green Mile.

That's difinatly The Green Mile http://www.imdb.com/title/tt0120689/

Yeah that's the "green mile", one of the best.

Yes the green mile

I always wondered if Stephen King was influenced by "The Asphyx" (1972) with its immortal man and his equally immortal guinea pig! It's rather coincidental.

https://en.wikipedia.org/wiki/The_Asphyx