Horses get sick because it is so good they eat a horribly large amount and then they colic. Which is essentially their intestines twisting and bloating and they are unable to pass it all. I am very doubtful chickens are large enough for this to happen. At least I've never heard of colic-ing in chickens, but still doubtful since they have a different digestive tract, as their food gets broken up int he gizzard before it moves on.
If they're free ranging they don't need the grit. If you're going to confine them in the coop for an extended period of time, they'll need grit. They keep the pebbles, etc in their crop for a long time, so you don't have to give grit if they're locked up overnight or just for a few days.
Depending on how much clippings you have, I would make sure to spread them out some when you put them in the run. If you just dump them in a pile they can heat up and mold. Although, if your chickens are right there, they'll spread the pile out for you!