So, a second chick had the same thing! Except it was one of the Swedish, and it was an even bigger piece. We got it out of his throat, packed up the chicks, and moved them to their brooder in the barn. If they hadn't of done this, they would have had another week inside.
We used a plastic tarp, like what you put down when you paint, to cover the bottom of the dog crate. They just got too curious, as chicks tend to do, but luckily we were prepared with their outdoor home.
I've alwasy said, "Chickens are pretty smart for being so dumb"...today was a dumb day. Now we have to readjust the brooder, so this doesn't happen again. Although, it'll start being much warmer very soon, and maybe we'll just brood in the barn from day 1. I just love how friendly they are when they've spent time with us.