CK is an employee of a private corporation. Based on the Brady appeals court ruling, Goodell has unlimited power to suspend players. Not that Goodell will risk suspensions. It comes down to balancing public relations of outraged fans, and the risk of more players taking a similar stand. Which is starting to spread by way.
Your analogy falls far short, because you are comparing public schools to private corporations. Any one working in a private company can say what they want, and supposedly won't be jailed. No protections from a disagreeing employer firing an employee. Courts have already ruled that public school students are not required to pledge allegiance. But maybe you are not current with court rulings.