Assuming you already unlocked the exclusve egg on both devices and have a slot to hatch an egg:
You will need to backup your save file using an eprom flasher, flahs this save file to the device you want the exclusive egg, press A and B then select the exclusive egg.
Ass soon as it shows the egg pulsing on your screen, you have to take the battery off so it wont hatch.
Make a backup of this new save file with the exclusive egg and flash it back to the original device and youre done.
