The short answer is that the First Amendment doesn't apply to private employers. But, that is not the end of the story in the NFL case.
The NFL is a union shop. All players of union members and are subject to a collective bargaining agreement with the owners, in addition to having individualized contracts with their respective franchises.
Generally speaking, one of the core provisions of any collective bargaining agreement is to prohibit employees from being fired without cause during the term of their employment contracts. Union employees are almost never employees at will. And, the collective bargaining agreement will generally have hard fought definitions concerning what constitutes good cause to fire an employee and what does not.
So, while the First Amendment doesn't apply to the NFL, that doesn't mean that NFL players are without all protections from wrongful firing. But, in the NFL case, this turns on the definition of "good cause" adopted in a negotiated collective bargaining agreement between players and owners, and not upon the United States Constitution, and is resolved via an arbitration process agreed to between the union and the owners, rather than in the court system.
Further, as @bdb484 notes, state laws sometimes prohibit firing employees for particular reasons, and specify the consequences of violating those laws.
For example, Colorado law, which would presumably apply to individual employment contracts within the collective bargaining agreement between players for the Denver Broncos and the owners of that franchise, prohibits firing employees for off duty activities that don't affect their work duties subject to certain exceptions, with specified penalties if this requirement is violated (that are generally less generous than a wrongful termination suit, for example, alleging racial discrimination).
This particular Colorado law wouldn't be relevant to something that an NFL player does on the field, but might impact the ability of the NFL to fire a Denver Bronco player for some kinds of off the field conduct. (The statute was enacted in the first place, primarily to protect employees from being fired for smoking tobacco while away from work.)