I wouldn't bother with assembler since PBP has WRITE. That should be all you need.

WREN is clear on power-up so I'm not sure how spurious writes would happen at POR.

You could clear EECON1 at power-up with EECON1 = 0 just in case.

This 18F part has a high/low voltage detect, comparators, and several settings for the
brown-out-reset voltage, so you have quite a few options.

If you have a spare input, and can use it to allow EEPROM writes only when a button is
pressed, it would be a good test. Maybe even blink an LED before & after an EEPROM
write. Then you could monitor the LED to see where the un-intended write was taking
place.