No, eggs should not be washed before refrigeration. Washing removes the protective bloom, making them more susceptible to bacterial contamination.
In the United States, eggs are commercially washed and sanitized before being sold. This process removes dirt and debris but also removes the natural protective coating called the 'bloom' or 'cuticle.' To compensate for the loss of the bloom, eggs are then sprayed with a light mineral oil to seal the pores and prevent bacteria from entering. Washing eggs at home after they've already been processed commercially can damage this artificial barrier, increasing the risk of Salmonella contamination. If you purchase eggs directly from a farm where they haven't been washed, it's best to leave them unwashed until just before you use them. Washing them immediately removes the natural bloom and leaves them vulnerable to bacteria during storage. Only wash them right before cooking.
If you find your eggs are dirty, gently wipe them clean with a dry cloth before storing them in the refrigerator. Avoid using water unless you plan to use the egg immediately.