How can I mount a light to a square junction box?
...ever seen lights that use this kind of box for hanging?


No, because that box is not made for directly attaching anything to it. The box needs an adapter ring attached to hang a standard fixture (or to mount switches, receptacles, etc.)

By itself that is just a junction box called a "4-S" (for 4" square). Look for a 4-S to 4-O plaster ring.

 

It goes without saying that it should not be sticking out of the ceiling like that.