To me (and keep in mind that I am by no means an expert) it looks kind of like an egg gone horribly wrong. Like the membrane formed, but the shell didn't. Although I have no explaination for the stench or where the yould would be. Unless the egg was stuck for a while and had started to rot. I've never seen or heard of it in birds, but I do know that snake eggs can actually start rotting before they are laid and when the snake does lay them they do not look like normal eggs (slimy, off color, etc).
Oh, just had a thought! Could it be possible that there is a snake or turtle nest nearby and one of the girls unearthed an egg and dragged it into the coop? It's a really really long shot, but the pics you posted do very much remind me of a broken reptile egg.