cattledog:
If you have a RTC (real time clock) module available, you might find that it has an eeprom included (typically 4K bytes) as part of the module. This will be an "external" eeprom, and is addressed with the Wire library and the I2C protocol. Anything written to it will be persistent through a power shutdown and reboot.
Unfortunatly i don't have a RTC module available, but thanks keeping this in mind.