Yup, that's fowl pox, the dry kind. Not much you can do but let it run it's course, usually about 30 days, more or less. You can put iodine on the lesions to help dry them up. Careful around their eyes, dont let any in their eyes, as you know it's very runny. Antibiotics are only good for secondary infections. Fowl pox is a virus and antibiotics wont help.
To control mosquitos you can hang sticky fly strips around the coop, a bug zapper helps alot and they cover about an acre or more. The eggs will be safe to be set on. Consider purchasing fowl pox vaccine for your chicks or chickens that havnt been infected yet. You can go to Jefferslivestock.com and read up on the vaccines. Just type in "fowl pox vaccine" in the search box.
Eggs are safe to eat.