The short answer is “No”. Eggs are laid with a natural coating on the shell that is applied as the last step in the laying process called the “bloom” or sometimes the “cuticle”. This coating is the first line of defense in keeping air and bacteria out of the egg. Since eggshells are porous, if you wash your eggs as soon as you collect them, you are removing that natural barrier.
Unwashed eggs can be collected and then left out on your kitchen counter at room temperature for several weeks, where they will still be perfectly edible, if not quite as fresh as when they were laid. In a perfect world, all of your chickens’ eggs would be as clean as these I collected from the nesting boxes.
There are some things you can do to ensure clean eggs, but since life happens no matter what you do, you will occasionally end up with dirty eggs – covered in mud or feces. You can gently scrub off the debris with a rough cloth or your fingernail, but extremely dirty eggs should be rinsed under warm running water and then refrigerated.
Once an egg has been rinsed off and had the bloom removed, it does need refrigeration. Unwashed eggs benefit from being refrigerated as well. In the refrigerator, unwashed eggs will still last longer than washed eggs, and also longer than they would at room temperature. For more information on proper egg handling and storage, read HERE.
Store bought eggs have been cleaned and the bloom has been removed, so always need refrigeration after you buy them. If you sell your eggs, you might want to read this – different states have different egg laws that govern whether eggs should be washed or not before selling them.