How can I seal this hanging electrical outlet in my horse barn?
Silicone caulk works wonders in all kinds of situations. The screws that are missing to hold all parts together can usually be replaced at any Home Depot or electrical supply store for just a few cents per ea. The hanging outlet box I have seen used several times in barns, however if a 2 by 4 can be attached to a  ceiling crossbeam and placed perpendicular to those beams it will offer more stability to the box and wiring. It can be installed high-up still rendering it useful and out of the way of any horses but will provide more stability to the whole thing.