When my girls lay their eggs in a dirty nest, I use a paper towel to brush the dried poop off.  If that doesn't work, eggs can be washed (carefully!) in warm water and that will usually take everything off. 

I don't resort to that in the beginning just because it also washes off the "bloom" on the eggshell which helps the egg stay fresher, longer.  The ones you buy at the store are almost always washed before packaging — eggs sold on the farm or by low-volume producers don't always have to be washed first.  Depends on state regs.

The reason the egg has the "bloom" is so when the hen poops out the egg the covering on the egg will protect the eater of the egg from Samonella. Most people who have chickens and wash their eggs do it without knowing this. That is why the egg wash you can buy by the gallon is so important to use if you sell eggs. Otherwise you should scrub the poop, mud, etc off before you sell the eggs. If you don't and you wash them you are washing the "Bloom" off and the egg is unprotected from samonella. So, wash with egg wash or use sandpaper or something you can get the stuff off with.

I find that it's better to scramble the eggs and then pour them into the ice trays.  My first year of freezing I simply cracked the eggs into the ice trays.  I found that when I thawed them that it was very hard to break up the yoke smoothly.  This year I scrambled and it's ever so easy to just thaw and use.  Just wish my ice tray cubes were large enough to hold a whole egg!

Oh, after they freeze in the tray I pop them out and put them in a freezer bag to store in the freezer until needed.