That is a real hard one to answer. It depends on how dirty they are and how and when that dirt got there. When a hen lays an egg, the last thing she does is put a coating on it we call bloom. That’s what makes a newly laid egg look wet. It quickly dries and forms a barrier that is pretty good about keeping bacteria out. It’s not perfect but it is pretty good.
If the ”dirt” was put on before the bloom dried, it’s pretty likely the bloom was compromised and you have a great path for bacteria to get inside. If that dirt is poop, there’s even more danger but thick dirt or something can be a problem. If it is a thin streak probably put on after the bloom dried, it’s probably not much of an issue. If it is a thick clump, even if the bloom dried first, there is more danger. It’s not an easy call.
If bacteria is not present they cannot enter the egg even if the bloom is gone. It sounds like you incubator is new. I’d still sterilize it, probably with a weak bleach solution then let it air out. You can remove the dirt by either washing the eggs or sandpapering it off. Either way will remove the bloom. If it’s not too bad you can try brushing it off. Whichever way you go, take great care to keep everything clean. Wash and dry your hands. Don’t put the eggs on anything dirty. Keep the environment as sterile as you can.
Then, on a real regular basis, sniff the eggs individually after you set them. If you detect a rotten egg smell, get that egg out of there before it explodes or leaks stuff that infects the other eggs. In an ideal situation I’d say don’t set dirty eggs but you don’t have that. You have a situation you need to deal with.
If you were using a broody hen I’d be a lot less optimistic because she is not a sterile environment. I can’t give you any guarantees but I think you have a decent chance of getting some chicks.
Good luck!
Sourland makes a very good point I did not think of. A hen should remain fertile two weeks or so after a mating.
