There are also laws in the US that consider all work done outside an employee's regular work day to be 'overtime', and which require compensation at a higher rate. This is true whether the work is ordered or not; these provisions of the Fair Labor Standards Act don't apply to everyone, but they would to the person in this example. A 'don't work at your desk during lunch' rule is directed at this, not at a clean office.

Regardless, firing for a single instance of this and then mouthing off at the personnel director isn't (IMHO) a firing offense, assuming that the media account is entirely accurate. [And, in the interests of full disclosure, as the saying goes, I used to be a personnel director.]