"The measure was inspired by a case out of Oklahoma in 2002, when a dozen paper mill workers were fired after bosses found out they had guns in their cars."
The companies asserted a right to conduct health and welfare inspections because the employees' vehicles were parked on the company's parking lot.
My recollection is that the parking lot in question was a leased lot shared by, perhaps, more than one company. I don't think the employees in question were at all aware that they were committing a firing offense for having their hunting weapons in the parking lot.
My solution to this problem is to make a change in incorporation laws.
Laws permitting the formation of corporations encourage the pooling of capital to conduct business by limiting the liability of stockholders. This is a benefit conveyed by law to improve life for all. There is no reason whatever not to include protections for civil rights as part of incorporation law.
Privately held businesses could continue to make any rules they want. It's unlikely that privately held businesses would be very interested in limiting the rights of employees in ways not practiced by larger companies.
There would certainly be little sympathy for an employer who banned political bumper stickers in their parking lots or who required voting a certain way as a condition for employment.
Liability concerns have resulted in very few larger privately held companies. Similarly, liability concerns cause corporations to decide issues such as "guns in the parking lot" with no weight whatever to the civil rights of employees. If there was no law prohibiting corporations from demanding specific political affiliation as a condition of employment, that would be viewed by most as a serious problem. Likewise, the keeping and bearing of arms is being made very impractical for most people.