If you were looking for something larger (say 21x21-ish) I'd say try to buy an old open side plastic dishwasher rack from a restaurant, then nock the bottoms out between dividers. Even new, they are only $10 or $11, but after a few thousand cycles they start to break from rough handling - places are always recycling them.
At 1' square, give or take, I don't have a similar idea, sorry.
You could buy a large, thick, plastic, round plant tray then drill a bunch of holes in it to make space for eggs, rotate it about a central point - but I'm not sure that's what you are looking for either, and if its not thick enough, it won't push the eggs well. What I've been seeing lately at the stores has (largely) been rather flimsy.