Ahh I see... I was thinking like a spoiled British passport holder...
When I drove to Canada, the Canadian border officials were OK with only the GC, and I'm pretty sure that's the routine. I believe if you have a GC in hand you don't need a passport to enter Canada or Mexico. The US CBP will also be OK with only the GC at re-entry.
Travelling through airport borders is a completely different beast through. I'm pretty sure they ask for a passport when entering the other country and also when re-entering the US.
At jfk I was specifically asked to show my passport. Also, the Mexicans will probably want to see your passport...