I use a piece of kitchen roll or a handful of straw to get poop of broody eggs. Whatever you use to clean them should be dry.
She shouldn't be pooping on the eggs though.
Is the broody hen getting off the eggs daily? if you're not sure lift her off the eggs yourself, make sure she eats and drinks and has a poop You might need to encourage her to leave the nest. Up to about half an hour away from the nest is no problem for the eggs.