Jennie, I googled cutaneous horns. I don't think that's it. This is the actual nail.

It's really tight together, but reminds me of this:
http://abcnews.go.com/2020/popup?id=3897859

It's really long and spiralled/circled into itself almost like a snail shell. I think the nail just didn't stop growing like the others have.

I'm confident the new vet clinic will get it taken care of!