There are many better ways to handle this sort of annoyance than to kill the dog in question. It might take some time but they could at least make an effort to try to find out who the dog belongs to or to create a situation where the dogs cannot steal the eggs. There are plenty of other creatures that steal eggs as well and this is the reason why chicken farmers have created coops for the chickens to keep raccoons, foxes, snakes, and other things away from them. Dogs are far more reasonable creatures than any of the other ones, so why not be a bit more compassionate to them as well?