It would still be "minds", because "everybody" is referring to a plural subject. You can say "everyone put on their jackets" but not "everyone put on their jacket". The latter makes sense if the listener dismisses the absurd possibility that everyone is fitting into one jacket, but it's not correct.
While 'everyone' sounds like it refers to a plural subject, it technically is singular. Therefore, "Everyone put on their jacket." is correct. This is another one where I would rewrite the whole thing to make it sounds better:
