Don't know if you have these problems up north, but here in Florida our mosquitos carry West Nile virus, malaria (it is making a come back I read) and encephalitis. The State uses chickens to test for the prevalence in regions across the state, so although I don't know if it makes the chickens sick, obviously they can harbor the viruses which could then be more easily transmitted to you...
The pesky mosquitoes can reproduce even in the bit of water held in a curled leaf on the ground: run a mulching mower under your trees and shrubs. Do make sure you get rid of water that accumulates in stuff in your yard. The tray under a planter can even be a breeding habitat. Take a good look around after you have rain to look for accumulation spots and standing water, then take measures to get rid of them. If your chicken watered is open, put screen over the top.
Also, the mature mosquitoes like to hide in dark places protected from wind--like in shrubbery. (This is why they are attracted to your coop, too.) Encouraging natural predators in that type of area in your yard is good, especially if it is where you chickens don't go: harmless spiders, or a bird box for your local species of mosquito-eating bird, for example, can serve as natural controls to keep the population down.
I have a dog run off my patio--I hung screen down from both edges of the 2x4 over the opening. Took a day to get the dogs used to it using treats, so I expect the same would go for the chickens. A few bugs get lucky and find their way in by chance, but it is the difference between 2 or 3 and dozens...
If you leave the human door of the run opn a long time you can also hang screen, but it will get blown by wind a lot so you might want to weight it or use Velcro to keep it more secure. Velcro will need to be replaced over time.
I am in the process of building my coop, and the entire run and coop will be screened in: I'm using aluminum screen down low, and the cheaper nylon screen up high.
Since I want to let the chickens free range in the day, I'm making a chicken door in it so I can leave the human door shut. It will have 2x4's all around (on edge) separated by 3/4" shims with a slot for a slide-in 1/2 plywood door to shut at night, with screen hung off both outside edges.