The expression commonly used in both the affirmative and negative. The indefinite pronoun 'everyone' is a singular pronoun. There are actually two issues concerning this topic:
Your deduction, that not everybody's cup of tea means not to everyone's liking, is absolutely correct. Joe got everyone's attention and started to speak. Should it be everyone's, everyones' or everyones?
In my experience, the spelling checkers get confused by contractions. Everyone who is coming will receive a gift. An example would be it flew over everyone's heads, or it flew over everyone's head. May i have everyone's attention,.
Would the noun following everyone's be plural? I have the following sentence: Are the words everyone and everybody singular or plural? The possessive of everyone is everyone's, in the same way the possessive of everybody is everybody's.
What would be correct in this case and why? Unless of course you’re a buddhist or a cat, in which case everyone has multiple lives and both would be correct, depending on context. There is a comprehensive article on the topic on grammar girl: In fact, to catch someone's eye is an idiom, not eyes. the question is not about everyone being singular or plural.