As said before..prevention is worth a pound...
I find that using the deep litter method of bedding really tends to clean off the debris of muddy feet before girls get into the nests. Keeping the bedding clean and dry is also a plus. Also, you might look into if you have enough nest boxes to provide less crowding. The less birds getting into each nest, the less debris tracked onto the eggs already there.
Also, if your eggs have a lot of manure residue, you might look into what you are feeding. Most chicken manure should be a healthy, firm blob that drops from the vent and doesn't adhere to the feathers surrounding the vent. If your chickens are running around with wet vent feathers you may either be feeding the wrong feeds or your run is too wet and muddy.
I keep a hay bale in the building next door and, when I gather eggs and find a soiled nest, I simply flip that out into the bedding on the floor and grab a fresh handful to refresh the nest. It only takes a matter of seconds to put fresh nesting material in.
When I find soiled eggs that have dried matter that doesn't wipe off easily with a damp cloth, I either use that one for supplemental protein on the dog's food or will return it to the manufacturers. If I really need that egg to fill an egg order, I will brush the debris lightly with a dry scratcher and then wipe it gently with a damp cloth.
