Hi, 

Foxes are the most likely culprits, but raccoons, skunks, or possibly rodents (rats) may do this as well, storing them for later. They have probably identified your yard as a safe area, with no competition or other predators to take them. 

There isn't a way to tell for sure beyond catching them in the act with a wildlife camera or spotting them in the night. 

There probably isn't a way to deter them from doing this. The somewhat nearby neighbor would need to protect the eggs better. Are you sure it's chicken eggs? It could be a duck or another type of larger bird (over a songbird).

If they are in an area where they can stay, it'll be your choice to let nature run its course. If they are near your home or in a place where perhaps a mower would hit them, you may want to remove them, just for smell reasons, if you break them and they are rotten.

Emily