What kinds of flies congregate around outhouses during the spring? How do we get rid of them?

There are a few possibilities, but outhouse flies are most likely cluster flies or filth flies.

Cluster flies get into buildings in the fall, seeking warmth, through small cracks and crevices. They winter indoors, emerge inside in spring, and buzz around the windows trying to get out. Eventually, they’ll leave or die trying, but not before thoroughly annoying you with all that clustering. Because cluster flies can squeeze through the tiniest of openings, preventing them from getting inside structures is tough, but certainly worth a try. Seal any holes and cracks—especially around the windows—with caulk in the summer. Since you would never want to completely cut the airflow to your outhouse, cover the vents in your privy with standard window screening. Make sure it’s snugly fastened.

Filth flies vary in appearance—the term covers a wide variety of flies—but they’re all about the size of a housefly, and they all hang around garbage, manure, and dead things. You don’t want to get rid of these guys because they help to break down fecal matter, and, in the outhouse, that’s good. Try to live with them.