Horses can be specifically trained to be war animals. This includes a lot of very specific skills, including not shying at gun fire or other auditory or visual distractions, holding their urine until their saddle is off (horses must stand still to urinate, which could be deadly during an attack), and being quiet at all times. Part of this is temperament, and some horses simply aren't cut out for the job.

I doubt that the horses were kept from neighing in that specific instance-- it's more likely that they were trained from an early age to be silent. Horses are still being trained for cavalry today as a recreational activity, for example in Civil War Reenactments. Here's an interesting article about a girl who's trained her horses to be battle-ready.