"Everyone . . . their"
In general, pronouns should agree in number and gender with their antecedents. Speakers and writers of English most frequently depart from this rule when they sense that it would be inappropriate to limit the person picked out by the antecedent to a single gender. The problem here has long been recognized; the original edition of the Oxford English Dictionary noted matter-of-factly that the "pronoun referring to every one is often pl\[ural\]: the absence of a sing\[ular\] pron\[oun\] of common gender rendering this violation of grammatical concord sometimes necessary," and gave, among its examples, this sentence from 1877: Wiki Markup
Everyone then looked about them silently, in suspense and expectation.