It is not really possible to know for sure whether the company has committed an unfair labor practice without have a very detailed account of what the employee said on Facebook. It would be difficult even with all the details. Without details, it is almost impossible.
The basic rule of thumb is that employees cannot be fired for posting negative things about the company if they are involved in some sort of “concerted effort” to bring about change. This dates from a law from the 1930s that was meant to protect employees from being fired for union activity. However, if an employee is simply complaining about work conditions, there is no “concerted activity” and therefore the worker is not protected.
Of course, it is difficult to know when “concerted activity” is happening on a platform such as Facebook. There was a case, for example, where a bartender complained about a boss and a cook from the same bar clicked on the “Like” button on Facebook. This was seen as concerted activity. By contrast, there was a paramedic who was fired for calling her boss a “scumbag” on a Facebook thread in which various employees had been discussing their bosses. This was deemed to be mere complaining and the firing of the employee was upheld.
As these examples show, it is very hard to know whether an unfair labor practice has been committed.