Yes, the salon can ask you to do this. And unless you currently have a contract which guarantees your employement--e.g., a contract guarantying you the right to work there for, say, the next year--the salon can enforce their "request" by not employing you any longer if you won't sign the agreement. Any employer, whether of employees or of independent contractors, may make it a requirement of employment that the worker sign an non-competition agreement. Your recourse, if you don't want to sign it, is to stop working there. You also, of course, can try to negotiate different terms to the agreement, but like any negotiation, that is voluntary--i.e. you can't make the salon offer different terms, but only see if you can work something out with them.