I am not an English teacher, so if a teacher wishes to correct me, please do!
Everyone is a singular word with no plural. It means "each person" or "every person". So, just like these two word expressions, words associated with everyone, or everybody, should be singular. "Everyone's life" is therefore the correct answer.
Interestingly, "everyone's lives" actually sounds right to our ears. I suspect because "everyone" seems to be talking about the 6.8 billion people of the world, and that just cannot be singular!!