Progressive Discipline for Badmouthing the Company Online

One of our employees became disgruntled after her son was expelled from a school that, while not technically under the [name omitted] umbrella, is a sister institute. The employee has been posting negative comments about the school on Facebook. Even though she hasn’t criticized the [name omitted], she did mention it one of her posts and acknowledged the possibility of getting fired for doing so. We had previously talked to the employee about a different posting and she took it down. Her Principal then met with her in response to this recent post, i.e., the one mentioning the [name omitted]. But, so far at least, that post has not been taken down. We plan to meet with the employee and impose some kind of discipline, like a suspension for insubordination. Are we handling this the right way? And are there any legal implications that I need to be aware of?