Stop Buttons
The biggest and most important "stop button" training needs to always first be with us! We need to study, to learn our horse's individual stress, conflict, and avoidance signals and LISTEN to them! Listening to your horse's natural language is the first most important thing you can do to empower your horse to communicate more. Especially the small language. Large demonstrations of body language is like the horse screaming, so listen to the whispers, the ear twitch, the leg shifting, the tight lips. Listen!
This being said, there are scenarios where we may want to give our horse a specific way to communicate to us that they want us to stop. Maybe their usual way isn't a safe way for us, maybe they have learned dangerous behaviors work and small signs don't so they jump right to large displays of "STOP!" Or maybe our horses will be working with people who aren't adept at reading horse language but we can give them a clear rule like "when the horse touches this target you need to stop". Like horses working in therapy or kid's programs.
Stop buttons also need to be respected, if you are going to teach your horse a way to tell you to "stop" you need to listen. It's not fair to teach them a cue like this then ignore it, that's asking for trouble. So don't open a line of communication you aren't willing to listen to.
To teach a "stop button" choose a specific behavior which can be removed in "no choice scenarios" like the vet. Encourage the horse to do the behavior then respond with a similar reinforcer you would use with the situation had they said "yes". At the same time as removing anything related to the situation away from the horse. Practice this a few times for them to get the idea. Then ask for the "real" thing you want, if they choose the "stop" option, reinforce, stop, and go do something else for a few minutes before asking again.
If the horse understands the process and tells us "stop" we need to reinforce it similarly to what they would have gotten had they done what we asked, so as not to negatively punish the "stop" button. We can throw a big party and use higher value reinforcers for a "yes", but always reinforce the "stop" as well.
A simple example is giving a horse a bath.
While standing facing forward at the bathing area you begin spraying the horse with the hose. You reinforce periodically with hay pellets as the horse stands relaxed for hosing. The horse touches a specific target which means "stop". You immediately turn off the hose, give the horse a small handful of the hay pellets and go do something else with the horse for a little while. Then you return to hosing later and the horse stands calmly, you make a big party for the horse and reinforce heavily. But if they return to their stop button, we listen and accept that.