The only thing you can do is give the birds clean dry underfoot.
Start with a doormat! I used to find that bark chippings, a good thick layer, worked well to keep them up off the mud. And if there's a 'porch' in front of the pophole, so it doesn't get rained on directly, then you could put straw on top of the chippings too, which would clean their feet up even better.
Then clean carpets - a nice thick layer of fresh wood shavings inside the house for them to walk across en route to the next boxes.
And fresh hay in the nest boxes themselves.
Finally, collecting the eggs several times a day, to reduce the number of feet which have trampled each egg by the time it gets collected.
I'm sure you already know that washing eggs is a no-no, and if you do use water near eggs it needs to be warmer than the inside of the egg, or it washes any germs
into the egg through the porous shell.