Oddly enough, given they're herbivores, eggs are a traditional horse feed, most often given in the racing industry. Some commercial protein supplements also contain egg whites.
There are no reported ill effects from feeding horses eggs (some people say you should cook them first, but the tradition is to feed them raw - I wouldn't recommend that with commercial eggs, though, salmonella and all). They're most often fed to young horses that need extra protein, but are also supposed to be very good for the horse's coat. It's now become a somewhat controversial practice, but is still quite common in some places. If you're writing about racehorses in a time period other than very recently, an egg a day would be normal and it would be very strange not to do it.