Should NCOs and OICs be monitoring subordinates on FB?

I think we have more important things to do than snoop around our people

While anything you post online should be assumed to be 'public', I do not believe there is a reason to cyber stalk your soldiers. If there was a report of impropriety of some type, that would change the dynamic (as well as the response), but as a general rule, I would say no. Everyone is entitled to a degree of privacy in their lives.

Unless I were a member of a unit group/page of FB, frankly it is none of anyone's business but my own, and I as a leader would never have even considered attempting to monitor my Soldiers' Fb activities without any outright evidence of wrong doing that would be detrimental to the good order and discipline of the unit and/or Army.