Gosh, Jackie, I never thought about it. My grandma just washed them with soap and water and dried they with a cotton dish towel, usually an old flour sack. Washing certainly didn't seem to hurt them. She washed them if they had poop on them or if she planned to boil them. Otherwise, I don't recall her washing them before she broke them in the skillet to fry or put in a recipe, but she may have. I think a special cloth would be a money-making gimmick for the person who marketed it. (Kinda like baby wipes, ain't nuthin' better than soap and water, mama.)
On another note, I think I've seen some instructions to cleanse boiled eggs with vinegar water if you are going to dye them. When I was a child, I think our dyes came with instructions to add vinegar to the dye, so we just made sure our eggs were clean and boiled.