Ok Lightning,
I rescued some muddy eggs before by washing in Betadine and hot, hot water.  The water must be 100 to 110 degrees Fahrenheit.  Too cold and you'll have gunk enter through the pores in the egg.
1.  Add some Betadine (Povidone-Iodine Solution, 10%) to a clean plastic shoebox sized container.
2.  Fill the container half way with clean water (100 - 110 degrees F - I use a digital baby thermometer)  You may have to add hotter water or colder water after you test the temperature.
3.  The water should look like dark tea.  I didn't measure, I just poured.
4.  Put the dirty eggs in the water for one to three minutes.  Don't answer your phone or start another activity.  Stay with your eggs!  
5.  Take out the cleanest egg first and rinse it in the hottest water your hands (in gloves) can stand, about 115 degrees F.
6.  Put the rinsed eggs on a clean towel to air dry.
7.  Repeat step 5 until all your eggs are rinsed.
There's a website out there somewhere with more specific information about washing eggs using Betadine solution.  I can't find it.
I figure if you're going to toss the eggs anyway, why not try a hot, hot Betadine wash.  Cold - Warm water gets into the egg.  100 degrees + doesn't pass through the pores.  Over 120 degrees and you've poached your eggs.