I took @aart suggestion, who covers the nest boxes in the late afternoon before they go in to roost, then uncovers them at night after they've gone to sleep. I was just covering them at night and leaving them until morning, when they were looking to start the day's laying, but it made so much more sense to have them already open when they wake up because I have a couple of very early layers and I am basically a lazy, sleep-in kind of person.
I know I can't prevent every poopy egg. The Brahmas, with their big bodies and feathered legs, do tend to drag stuff into the nests on their feet. When I integrated the young chicks with the older birds, the younger ones would often just sit in the nests watching the Bigs with great suspicion and pooping if they felt the urge. It took a lot of grabbing and tossing to break that habit. There has been a dramatic reduction in the amount of poop on eggs, but now that it's just a couple on occasion it's pretty easy to just wipe those off.
I know I can't prevent every poopy egg. The Brahmas, with their big bodies and feathered legs, do tend to drag stuff into the nests on their feet. When I integrated the young chicks with the older birds, the younger ones would often just sit in the nests watching the Bigs with great suspicion and pooping if they felt the urge. It took a lot of grabbing and tossing to break that habit. There has been a dramatic reduction in the amount of poop on eggs, but now that it's just a couple on occasion it's pretty easy to just wipe those off.