“Everyone must sit in their own chair!” (Shudder!)
Everyone means “every one.” “One” is singular; in fact, it is perhaps the most singular of all words. “Their” is plural. “Their” and “one” do not match. Grammatically, “one” takes “his,” “her,” or “its” — never “their.” So the correct expression is “Everyone must sit in his own chair!” (or “her own chair” or “its own chair”).
If political correctness requires gender neutrality, try “his or her own chair.” (Under the traditional rules of English, “his” does not refer exclusively to males, but rather to members of the human race of either gender (and, by extension, to a few other species — horses and dogs for example). This rule was repealed, apparently, during the Women’s Liberation movement of the late 1960’s and early 1970’s, resulting in a more complicated solution to the problem of being all-inclusive when using a singular possessive pronoun.
In most languages in which all nouns have gender as a matter of grammar, this problem does not arise: You can be gender-inclusive without using a plural pronoun with a singular noun. This is because possessive pronouns modify the nouns they describe, and, when necessary, they agree with the gender of those nouns. The possessive pronouns in other Western languages do not imply anything about the gender of their owner or user.