I have had four horses, with lots of whole and broken clay birds around, and I never had a problem with horses eating them or getting sick. In short, I don't believe that a horse will ever eat them.
Pigs, however, will eat them. Also, I've never seen anything that a goat won't at least chew on. One time, I gave a goat some bubble gum to see how long he would chew it, but he kept spitting it out and them picking it up again. After it got totally covered with dirt, I finally threw the gum away. I believe that the goat would have just kept chewing on it, and spitting it out, and picking it up again all day and night.
Best Regards,
Big Paulie