It's not about the airline, but the ground handling agent. All of them are supposed to use IATA's TIMATIC database to verify documents, which states:
Passport Exemptions: - Passengers with a Permanent Resident/Resident Alien Card (Form I-551).
However, you may well have to somehow force the staff to actually check TIMATIC, as many lazy employees will instead go by their own personal presumptions about what the rules are, with its consequences.
I'd recommend you to clear this up with the respective handler in advance, just to be safe.
For Jetblue the handler at Cancun is Swissport, whom you can email at firstname.lastname@example.org or call at +52 99 8886 0764/65/66
Also, your driving licence is irrelevant; no need to show it to anyone.