Yes, it is correct, but it doesn’t use parallel construction. If we say “everyone’s” then we should follow it up with “no one’s.” Conversely, if we use “everybody’s” then we should follow it up with “nobody’s.”
Make it either “Everyone’s friend is no one’s friend” or “Everybody’s friend is nobody’s friend.”
Make it either “Everyone’s friend is no one’s friend” or “Everybody’s friend is nobody’s friend.”