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."