Can horses get diarrhea?

Unfortunately, yes. I've talked before about colic, which is often caused by constipation. Horses can get diarrhea, from any number of causes.

They can get diarrhea from a bacterial infection or overgrowth (stomach flu), parasites, inflammatory bowel disease, antibiotics, stress, a change in feed or eating something toxic. Some horses are prone to chronic gut imbalances and may need to be regularly fed probiotics to stop them getting the, shall we say, trots. In some cases, there's no way to actually cure the diarrhea, just make it less severe (It's rare).

It's not uncommon for horses to get diarrhea for a couple of days right after moving to a new barn, especially if you have to change their feed.