Yes, mosquitos carry the virus, but it is also transferable through the direct contact of the birds, contaminated feeders/waterers, and possibly a few other ways.
You shouldn't eat the eggs if you have been treating with antibiotics. Fowl pox is not transferable to humans.
If they get them around the eyes/mouth (*and it is interfering with them getting enough to eat) then once a scab forms, I would just gently try to scrape them off. They need to be crusty - (I hate that word), otherwise it will bleed.
I've read that you can use a q-tip with betadine to help the sores heal faster/prefent further infection. I'm not sure if this helps or not, but it couldn't hurt.