They certainly shouldn't get sick, but as
@Chookchicken mentioned, watch out for that egg eating, you'll be pretty unhappy if they start to destroy your eggs on you. Are they on a layer pellet or have access to oyster shells for calcium? If there's no one able to collect eggs during the day, a few goofballs or ceramic eggs (most farm supply stores sell them, so does Amazon) in the nesting boxes might be enough of a deterrent, but egg eating can be a tough habit to break if it happens once too often. There are a few other tricks you can use before throwing in the towel though, so be sure to mention it if it starts to become a problem.