i am a hairstylist and have recently moved from one salon to another i had placed a sign in the salon window that said my name and that i was formerly the other salon is it illegal for me to use that businesses name with out there permission can they sue me if they wanted to

Unless you agreed via contract not to use the former salon's name, then I see no problem letting potential customers know who you are and where you used to work.  Your prior employment history is a simple fact that would appear on your resume and, all else equal, I see nothing illegal or actionable about that. Where people can get themselves into trouble is by: (i) publicly making disparaging, untrue comments about former employers...that's a different ballgame altogether; or (ii) illegally copying or using (infringing) a former employer's tradename, logo, or other intellectual property for the purpose of confusing or misleading consumers as to the real source of a product or service.