A policy is great to help, however I think DeeQ was on the right track. If you notice this, instead of flaming the employee for being displeased, you may want to consider uncovering the root cause of why they not ONCE but TWICE said bad things about your bank online.
I personally know tons of people who tweet and post things like... "TOTALLY NEED A DRINK after that manager's meeting." Or "Dreading going to work today..." While not generally meaning more than the general aggravations of work life... there may be a larger undercurrent of discontent that you have been alerted to. In many cases it may be just a matter of time before the undercurrent becomes a tsunami of discontent that starts to impact the effectiveness of the organization. You should be researching SM posts not only for "inappropriate use" such as saying what a terrible place your bank is to work... but figuring out why someone things your bank is so horrible... don't be afraid of actually doing HR things like investigating employee complaints, just because they come through Social Media... it is just a new outlet for venting and in some cases as the NLRB link above points out, it may be legally protected.
Understand you can't please everyone, but one person can really spoil things for everyone else. If you respond to customers who blast you on SM with top notch service to avoid bad publicity, why is it taboo to use the same customer service approach to respond the same way to employees who do the same??
Cheers!
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In life, there is a lot less that could get better and a lot more that could get worse.
MBA Fin/MBS HR
My views only!