When it comes to signal range a satellite is functionally a REALLY tall tower, nothing terribly magical about it, and still subject to being on the other wrong side of the horizon and unreachable. If the satellites your GPS is talking to are exclusively over America you will still wind up over the horizon eventually.
The signal loss in Mexico is persistent and continuous.
GPS satellites are not in geosynchronous orbit. The constellation is designed so that at least 6 satellites are in view from almost anywhere on earth and those 6 are constantly changing. You need to see 4 for the system to work.