Not unless:
1. They are constantly bringing it up at work and arguing with other employees and it's interfering with work getting done.
2. They are convicted of a crime in the course of expressing their political views (ex. the Capitol Riot).
3. They are a person whose public image is directly related to that of the company (ex. the CEO, a spokesperson, or other high-level person).
what about a cashier wearing a pro-Klan* pin on their lapel? If you talk to customers as a representative of the company, the company should (and does, as far as I know) be able to limit what political positions you can display while "on the clock". I suppose this could fit under your #3, but your examples seem to indicate you only want to punish a...ahem... certain type of person.
Also with #2, lots of people who got arrested at BLM protests/riots could be fired. Are you ok with that? Not that I disagree, if a person gains notoriety for doing an illegal thing during a political protest, they should be legally able to be fired for the negativity they bring the company.
*or antifa, MAGA, Socialists of America, the DNC, pro-life,pro-choice or any other political positions
1. The store could forbid wearing any kind of pins or buttons.
2. I said convicted, not arrested.
1.and indeed that is what most employers do (no political displays at work), and they'll fire you if you refuse to comply. This is how it should be everywhere that doesn't want to just do business with people of a specific political stripe.
2.more than a few were convicted to, you'd be ok with them being fired?