Hiccups - yes and no. They can get something called "thumps" - which looks and sounds like human hiccups, but is physiologically different. I think thumps is a spasmodic muscle contraction that gives a twitch and "thump" sound - whereas human hiccups are digestion related. One of my horses gets "thumps" now and again - and it does look and sound incredible like hiccups! It can be caused by electrolyte loss (in horses) and in these cases can be serious - but mild attacks seem to happen for no apparent reason and pass very quickly (with about half an hour) with no apparent discomfort or distress to the horse.
Most horses have 4 chestnuts - one on each leg - but some breeds, especially Thoroughbreds, Arabs etc (finer horses) often have less.