You don't need to wash or refrigerate your eggs.
In fact, it's better to do neither.
Chickens naturally coat each egg with a protective layer as they are laid, called a 'bloom', which protects the eggs from bacteria. If you wash the eggs you remove this protective coating. Also, eggs are designed to not spoil for several months in the wild, without refrigeration. Just wipe off any poop with a slightly damp paper towel, and keep 'em on the counter in a nice little basket for easy display.
Here in America we have this bizarre obsession with sterilization and preservation. In Europe, most people keep eggs(and butter) on the counter, not in the fridge.
Personally, I have chosen my hens to produce a colorful basket(brown, dark brown, blue, and green), and I want to see those pretty little eggs! Why hide them in the fridge?