Skunks are definate egg eaters. They are nocturnal, and they can't climb, so roasting chickens are safe from them, but they will kill anything on the ground. They are omnivores.
If you have skunks, you should not feed your chickens table scraps in the evening because this will attract skunks. Do not feed cats or dogs where there will be uneaten food within reach of them.
You can buy an animal trap, preferably an skunk trap. A skunk trap is enclosed on all sides but the front, which allows you to pick up the trap without getting spayed. Then remove the skunk to where you want to. I take them to the tailgate of my car and wrap a kitchen garbage bag around the front of the trap and the tailpipe, turn on the car, and they fall asleep and die peacefully in short order.
Don't forget, if you release them in the woods they will not have the abundence of food they find around human habitation and will either go hungry or return to bother people.