Private entities in the United States are more than well known for firing people for their political views and affiliations, Helen, and it is perfectly legal for private entities to do so.
That said, someone forgot to tell all the people pissed off about the NFL's supposed "disrespect" for the flag about US Flag Code Title 36, Chapter 10-176 C, D, or J, all of which the NFL and basically every other major sports organization in the country have been violating for decades:
(c) The flag should never be carried flat or horizontally, but always aloft and free.
(d) The flag should never be used as wearing apparel, bedding, or drapery. It should never be festooned, drawn back, nor up, in folds, but always allowed to fall free. Bunting of blue, white, and red, always arranged with the blue above, the white in the middle, and the red below, should be used for covering a speaker's desk, draping the front of the platform, and for decoration in general.
(j) No part of the flag should ever be used as a costume or athletic uniform. However, a flag patch may be affixed to the uniform of military personnel, firemen, policemen, and members of patriotic organizations. The flag represents a living country and is itself considered a living thing. Therefore, the lapel flag pin being a replica, should be worn on the left lapel near the heart.