Should be okay if the firmware is vaguely recent, and it's switched on (and not factory reset) close to to the rollover. Being on at the time probably helps. Dusting off your old yellow eTrex some time after the rollover might have issues.
I can imagine aviation GPSes having a much slower upgrade cycle than consumer handheld/cycling models. And 1999 was probably the first time anyone really thought about this...
It's 2038-01-19 that's going to be *really* interesting...