If you don't wash the eggs they will keep a very long time, I know I have kept eggs for
2 months but they would probably keep longer. Usually if I have any that make it to three weeks or more I feed them to the dogs.Eggs are sterile when the are laid(at least on the inside) and if they are not damaged or washed, will simply eventually dry out on the inside without ever rottening.
Chickens can and do eat anything, I had a couple of 4x8 sheets of blue foam board stored at the barn and the next time I saw them the only thing left was the thin plastic skin that is on either side. So much for organic farm raised eggs and chickens.