I don't know how big a body of water you are thinking of. Maybe the waterer the chicks use? Or a pond or birdbath? For small open areas like bird bath or a bucket of drinking water for other critters, you can put just a few drops of oil, olive or vegetable oil on the surface of the water. It will keep mosquitos from breeding. For larger areas, like decorative ponds, there is a little disc of mosquito killer you can put in the water. They are called "Dunks" or "Mosquito Dunks". They won't harm birds or pets, and dissolve slowly over time, I think. They make them for birdbaths, too.
If you think mosquitos are breeding in your waterer for your chicks, you can put a little piece of a Dunk in the container. They can be purchased at garden centers.
But, honestly, I don't think mosquitos would cause the kind of reaction you speak of.
Either an ongoing infection or something is irritating their eyes and nasal passages.
If getting rid of the lye and the shavings doesn't help. you could try an antibiotic ointment for eyes. Good luck!