Mosquitoes are attracted to that wonderful plume of CO2 from all those birds breathing, the heat from their bodies, and the moisture.
You say you don't have any standing water other than their drinker, but take a look around your place and your neighborhood. Depending on the species of mosquito (we have over 60 kinds here in Ohio, if you're in the south, you could have quite a few more!) they travel a few hundred feet or as much as five miles from the water they grew up in.
If you can find their breeding and larval habitat, get rid of it or treat it. This is a great feel good project for neighbors. When I did vector control, people were easy to convince that mosquitoes were bad, and easy to motivate to eliminate habitat.
I can offer a few ideas about discouraging mosquitoes around your flock. I've never been faced with the problem, so I don't know how well any one might work for your specific situation.
1) Go Old-School. Malaria control in developing countries often uses DDT (bad) or malathion (not really that good either) sprayed onto interior walls. This will work, and won't really cost much, but malathion is unpleasant stuff. Still, I'd consider this if the flock was under lots of stress from the mosquitoes, or there was a confirmed arboviral outbreak going on.
2) Exclusion. Screens and nets and such to keep the insects out. Those work well if you keep the birds in.
3) Mosquitoes have a hard time with wind. A 5mph breeze will keep them from flying. I don't know if this could be used to your advantage.
4) Give your neighbor a bug zapper, or a Mosquito-Deleto. These things do a great job of catching less than half the mosquitoes they call in from all over the world. Let your neighbor call them out of your coop and over to his porch.
5) Flypaper? Never tried it. Run an experiment and let us know!