They can fire you for any reason or no reason at all as long as it's not illegal discrimination (and it's not).
Then can object to your taking time off from your normal work day.
They can behave unprofessionally. Nothing illegal about that.
You are not required by law to find a replacement, but your employer is not required by law to continue your employment if you don't do as you are told.
The employer's rules and desires do not need to be in writing to be enforced.
If you get fired under these circumstances, you would probably qualify for unemployment.