The problem with everybody and everyone is that they are grammatically singular but notionally plural. Once it was said that a plural pronoun to refer back everyone and everybody is not considered a good writing and it should be avoided in serious writing or literature. Even this suggestion was opposed by then contemporary grammarians. In fact there had been disagreement among them. But it is noteworthy to mention that both singular and plural pronouns were in use to refer back "everyone* and everybody from a very long time, not only in casual informal speech but in literature as well.
... everybody had their due importance - Jane Austen, Mansfield Park, 1814
Experience is the name everyone gives to their mistakes - Oscar Wilde, Lady Windermere's Fan, 1893
Well, I'm afraid it's everyone for himself - R. Hall, 1978 (NZ)
The idiom would be one of solid understatement with everyone turning his hand to what needed to be done - N. Barley, 1983
In view of the above it is not wrong to use either "they" nor "he" in the tag question.
Everyone is here, isn't he?
Everyone is here, aren't they?
Just want to add something based on my opinion and findings that they is much more common here than he.
(I would not have added this answer here as a correct answer is already given unless OP asked me to provide him with some example sentences.)