Is it okay if I refuse to supervise another employee at another company?
Why do you want refuse to supervise that employee?
He will ruin my business and my reputation.
Even if you don't agree with the orders you are given at work, you should still try your best to comply with them. Refusing to do things that are your responsibility at work will result in you getting in trouble and jeopardizing your job. While it is understandable to have concerns regarding a problematic employee, you should think of it as a challenge and opportunity to improve your managerial skills.
I'm going to ask someone else to supervise him.
You need to at least try if that's what you were told to do. How would you feel if someone ignored an order from you and started doing whatever they wanted to?