As an initial matter, the NLRB does not defend "political speech." The NLRB defends employee action done in furtherance of "mutual aid and protection." Unless your advocacy has something to do with the terms and conditions of employment, your political speech is NOT protected in my opinion.
Because your employer is a private employer and not a public employer, I think they could terminate you for political speech.
You should talk with a reputable and knowledgeable lawyer.
Best of luck,
Howard B. Hoffman
There is no freedom of speech in the workplace; the only time that an employer cannot curtail your rights, is if the employer is a local, state, or federal government.
So short ans to your Q: Yes, they can terminate you for not following employer protocol/policy.