If we do, within minutes, there will be an egg on the floor, with everyone gathered around frantically lapping it up until we arrive. They also love them cooked. You've heard of egg-sucking dogs. Cats are notorious for sucking eggs in a henhouse, too. Sometimes a hen does not excuse herself to the ladies room before making a deposit in the nest, and sometimes an egg is broken and gets the other eggs in the nest eggy.
Some people keep a little piece of sandpaper handy to scuff off little dirty places.
Actually, the hen deposits a coating called the bloom on the egg which helps keep it fresher if it is not washed off. That's why we put clean eggs straight into the cartons and in the fridge. Hope this helps. Yes my cats also rellish any dropped eggs but as yet they do not 'egg suck' down at the chicken house. They put a tiny hole in the tip then spend ages sucking and licking out the insides.