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The possessive of everyone is everyone's, in the same way the possessive of everybody is everybody's.

In my experience, the spelling checkers get confused by contractions. The one that runs on the Mac OS X has problems when I write, for example, it's Sunday and it suggests me to use its if I meant to use the possessive; the previously used message was suggesting something different, showing that the spelling checker had doubts about what I meant with it's, even in a phrase similar to "it's a cat."