Could be a crow, though they often take the eggs with them and stash them elsewhere. It could also be your own chickens. I have had many chickens turn to egg eating over the years. Some of them eat the shells too, some don't. Recently, one of my miniature horses started eating eggs, she doesn't eat the shells, either!
Sounds like your chickens like their own eggs too much! Try taking a few whole eggs, blow out the insides so that you still have the shell intact, and fill the shell with salt. Then put them back in the nest. I just learned that trick from a very prominent breeder on BYC! You may want to mark the eggs so you don't get one of the saltys!
I agree with the other posts...sounds like you have a bird or two eating the eggs. I have never heard of that salt remedy but sure sounds interesting. Its really hard to break a bird from eating eggs once they start so I think it is worth a try.