Wow! Did I read that right...the company wanted to pay her to keep quiet about why she was fired?
And, as expected, the company's response was filled with, "blah blah blah, corporate speak, blah blah blah, corporate jargon, blah blah blah, corporate spin, blah blah blah, this is among the worst PR moves a company could make but we hope our corporate speak will hide that fact, blah blah blah."
Suspending her without pay for a week-ish and/or writing her up for breaking company rules might be appropriate, but not firing her (that is, unless she's been in trouble multiple times in the past and they were just waiting for a reason to fire her...but I highly doubt that's the case).