We have hairless dogs and hairless cats. Do we have any hairless horses?
The answer is: Sort of.
There is no harmless gene that causes hairlessness in equines, allowing the creation of a hairless breed. A hairless horse can result from the following:
1. Naked Foal Syndrome or Hairless Foal Syndrome. This is a condition associated with the Akhal-Teke breed (which is bred to have a very sparse coat). It's similar to JEB. The foals are born completely hairless and may have abnormal tooth growth. They have frequent digestive disorders, are highly prone to laminitis and almost never live past their second years. The vast majority die within a few weeks. (In equines, coat and hair problems are often linked to digestive problems).
2. Patchy hair loss is sometimes seen in horses kept in very hot weather, most often on the face or under the mane. This is harmless (if cosmetically annoying). Some horses will also shed out patches of coat before the new coat has grown in.
3. Skin infections such as ringworm. In very rare cases a skin infection can cause permanent hair loss (I've seen this once).
4. Any condition that causes itching can result in the horse rubbing hair off, especially the mane and tail. These are often related to insects or allergies. (Or allergies to insects). Mange is rare in horses, but does happen.
5. Selenium toxicity - caused by excessive selenium in the soil.
6. One specific breed, the American Bashkir Curly horse has a gene that when homozygous can cause loss of the mane and tail and patches of body hair in summer. (i.e., it sheds out and then never grows back in).
But there's no hairless horse breed. The only genetic cause of hairlessness in horses is a fatal recessive.