Manhattan Sentence Correction Guide, Chapter 2 , Pg 23.
Wrong: Does everyone have their book?
Right: Does everyone have his or her book?
I understand that "everyone" is singular and therefore "their" should be replace by "his or her".
However I believe with the same reasoning, "have" in the WRONG sentence should be replaced by "has".
Right: Does everyone has his or her book?
Is my understanding correct? If not kindly explain.