Not only will squirrels eat eggs, they'll eat baby birds. Not that many squirrels would want to tackle chicken eggs or chicks or take a chance on entering a coop. But they are omnivores & take advantage of high protein, high energy foods especially when they themselves have nestlings.
Fox squirrels are not nearly the nuisance that gray squirrels are. In my area, fox squirrels are becoming endangered as the grays ("city" squirrels) move in & take over their territory. The grays are the ones that we find in the garage, the ones that nibble on the door frames & destroy the bird feeders.
Occasionally, a gray will squeeze into the chickens' run (2x4 welded wire) & finish up any food they can find. So far, none has entered the coop.