I don't think there is much of a legal case here. But I can tell you from experience pressuring large companies over political email that you do have leverage. Companies like Google ("don't be evil") are very responsive to the threat of bad press. "Google is censoring political speech!" is a real bad press story and may get them to improve their spam-flagging system.
Cost of public relations/damage versus legal fees versus a quick $50,000 for signing off on a confidentiality agreement...basic tradeoff...seen it done many, many times...for a lot more money in some cases, too.
Yes...obeying an order to STFU has its price... <grin>
Some people actually make a living for agreeing not to do certain things...LOL!