Another way to think about it. Republicans are typically for smaller government. A non conpete clause is only enforceable via government petition. A private company cannot threaten force against a person because of who they work for. All enforcement is via government. Thus banning non competes leads to less litigation and a smaller government.
Finally there is no regulation being added to the market here. The government is in fact removing regulation by declaring via statute that it will no longer interfere when people violate certain types of agreements. Thats called the government butting out of peoples business
Lets not construct bogey men.