+0
If the subject is "Everyone", and you are talking about several people, would you use "they" or "his/her" later in the sentence?
Comments 
Depending on the case, use one of these: they/them/their/theirs
AnonymousIf the subject is "Everyone", and you are talking about several people, would you use "they" or "his/her" later in the sentence?

Because of the very nature of our language, AEmotion: star's answer may seem a bit confusing, but it's about the best one can do, since we don't have a gender-neutral pronoun in the singular.

I would just add a pet irritation that I encounter when "everyone" is used in the negative. All too often, we see "everyone isn't here yet, so we can't begin". To me, this reads that the place is empty. The better negative sentence would be "not everyone is here yet....".
Sorry but AlpheccaStars is incorrect because everyone is singular and the pronouns listed are plural.
You would use his/her if you wanted to avoid female reactions. His is appropriate as a generally inclusive term.