Dear Mulder,
"Right-to-work" is different from "at-will" employment.
"Right-to-work," as you point out, is about the status of unions and closed shops.
"At-will" employment means that your employer can fire you "at-will" for any reason at any time. And you can quit your job "at-will" for any reason at any time.
"At-will" employment is conditioned by the Civil Rights Act of 1964, and similar and related legislation, and by various state laws and judicial precedents.
Generally, exceptions to at-will doctrine are made at the federal level for race, religion, color, gender, and national origin. Also, there are certain limited exceptions made for dismissals that would violate "public policy." As an example, you can't fire someone because they refuse to commit a felony.
But in most states, you CAN fire people because: you don't like 'em; you disagree with their politics; they are homosexual (although a growing number of states forbid that); they refuse to follow work rules that don't impinge on the five protected categories mentioned in the Civil Rights Act of 1964, the Age Discrimination Act, the Americans with Disabilities Act, and the pregnancy protection law.
In any event, if an employer creates a rule barring possession of widgets on its property, properly informs all workers of the rule, and provides a clear process of discipline for violating that rule, even where employment at-will is weakest, the employer will generally be upheld. It doesn't matter whether "widgets" are guns or garbage cans, employers have a right to determine what will come on to their property and what will stay off their property.
Issues of enforcement get trickier.
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Also, there are certain limited exceptions made for dismissals that would violate "public policy."
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