First off - sounds like a terrible experience and sorry that you had to go through it

My take is you should get the $250, even if some of the delay was off the aircraft, but curious to see if there's a part I'm missing where all 6+ hours has to be on the plane. I've had a few instances where the system automatically sent me the wrong amount for the delay, I called JetBlue (1-800-538-2583) and they took care of it pretty quickly.

As for the 6-month expiration part, I unfortunately don't think they'll budge on that. What you can do, is just keep perpetually changing a flight booked with the credit after 6 months, but that's the best advice I have (see this thread for details on how to do it).