Questioning Equal opportunity employment.
It depends on the state, but here in Pa, they can fire you for whatever reason they like. If you think it's unjust, you have to prove it yourself. That's why I tell everyone to never sign a write up. If you do, and then try to say it was unjust, they'll say "well, you signed it, therefore you must have agreed it was wrong."
If you don't sign it, they will turn it into your file anyway, but when you complain about how you were fired, you can say argue your case more solidly.
They might have hired you with them, but didn't like them, or have gotten complaints from customers. They'll probably officially fire you for other little things you did or didn't do, but the underlying reason will be your hair.
NEVER SIGN A WRITE UP