Tom Cruise is a famous member of Scientology and we also had other Scientology-inspired movies in the past, like "Battlefield Earth". So is there any statement or evidence that "Edge of Tomorrow" is another "Scientology approved" vision of humanity and the future?
There is no relationship between Edge of Tomorrow and Scientology other than Tom Cruise's performance.
The film is based on a Japanese novel/manga called All You Need Is Kill.
Some sites claim that there is a correlation between Scientological dogma and the plot:
One more thing: if I didn't know any better, I might read Edge of Tomorrow as Cruise's secret Scientology movie.
But those similarities were already in the manga and weren't added to the script to make it fit their beliefs. Either way, you might find the article interesting.
Plus, according to Wikipedia, Cruise wasn't even the first choice to play the lead, Brad Pitt was.