I think all horses poop the minute they see the trailer. In fact, you can use this test if you think your horse might be collicking. So they may not poop in the show ring because it is all where your trailer was parked at home and in the trailer. But, I have even seen horses poop going over a jump.
However, my horses will not poop in their pasture. They all hold it until the minute they are back in their stalls. Or they will go poop in their run in stall if they can't hold it. People are always commenting on how there is no poop in my pasture and think I clean it everyday. But, the truth is the horses won't poop in it. Instead of cleaning my pasture I am always cleaning it out of their stalls. I think it is because the grass is food and they don't want that in their food. Some years ago I was given an old horse that was also blind. The first day she was here and went to poop in the pasture, the boss mare ran over and herded her over to the run in stall. After that she also started only pooping in there. Every horse here has learned to do that. Since other horses poop in their pastures though I wonder if my boss mare is responsible for my clean pasture. She doesn't want it there so no one else is allowed to put it there either. They don't poop when I ride them there either. I can't take responsibility for "training" them not to poop though. Given the choice I think I'd rather clean the pasture than the stalls!
Yes, horses can be housebroken apparently. Many disabled people are using minis as guide animals.