Most likely a snake as others have posted.
Please don't do the salt in the egg thing...snakes are beneficial and will take care of mice and rats just fine!
Snakes will eat whole eggs and as many as they can gulp down in a sitting, we have yellow rat snakes here in Florida...last one I got had six eggs lined up like six bumps in an inner tube!....when stressed they will attempt to regurgitate and escape but often will be unable to get through a chickenwire fence.
Depending on your situation, it is more than likely that the snake is hanging around waiting for the next easy meal and is somewhere around your coop. Spend some time looking, get a nice stick to move bedding and check under the coop, check the roof/ceiling, check little nooks and crannies and you will soon find his layup and can remove him or her. You must take them at least a mile or even two away from your coop to insure they won't be back. Do be careful when you first find the snake, it could be poisonous depending on where you live, once you identify it you can safely handle it either by hand or with a snake grabber (hardware/feed stores) and put him in a bucket to transport.
Try looking under your bedding if you have hay or suchlike in the nest boxes...often they will curl up under hay to sleep and digest the eggs.
If you're not comfortable handling snakes, see if you have a local wildlife trapper or nuisance animal remover to help you.
Good luck!....get a picture when you find him....