(1) Everyone likes the cake.
(2) Everyone has a role to play.
Hardly would any competent speaker use a plural verb for everyone:
(3) *Everyone like the cake.
(4) *Everyone have a role to play.
And that's all well and good. The indefinite pronoun everyone is singular and there's little confusion or disagreement about that.
Trouble starts when a pronoun is involved:
(5) Has everyone finished their drinks? (Longman Dictionary of Contemporary English)
(6) Everyone tells us that they enjoyed the movie.
If everyone is singular, why do even competent speakers of English use the plural their or they to refer to it? Is any grammar rule broken when we use their or they for everyone?
Here is what Quirk et al. (221) wrote about they in relation to indefinite pronouns:
The pronoun they is commonly used as a 3rd person singular pronoun that is neutral between masculine and feminine. It is a convenient means of avoiding the dilemma of whether to use the he or she form. At one time restricted to informal usage, it is now increasingly accepted even in formal usage, especially in AmE.
Rather than use he in the unmarked sense or the clumsy he or she, many prefer to seek gender impartiality by using a plural form where possible in reference to the indefinite pronouns everyone, everybody, someone, somebody, anyone, anybody, no one, nobody:
Everyone thinks they have the answer. [1]
Has anybody brought their camera? [2]
...
In formal English, the tendency has been to use he as the unmarked form when the gender is not determined. The formal equivalent of [1], though increasingly ignored, is therefore:
Everyone thinks he has the answer. [1a]So if you were ever doubtful, you can now set your mind at ease and use they or their to refer to everyone.
However, because everyone and they are generally understood as singular and plural respectively, mixing them can sometimes raise eyebrows.
In formal writings, therefore, the more careful writers prefer to use his or the less sexist his or her:
(7) Has everyone finished his drink?
(8) Has everyone finished his or her drink?
To avoid the problematic everyone and they, a common approach is to rewrite by making the subject plural:
(9) All of them tell us that they enjoyed the movie.
