Although you don't NEED to print out anything related to the ETA, I would strongly suggest that you do bring along a printed copy of your confirmation with you. Always better safe than sorry.

At least that was the case with my NZ flight last October - LAX-AKL-MEL. Upon checking in at LAX, there seemed to be some problem with my ETA and they were about to deny me boarding. I think some computer glitch had transposed the numbers or something like that - not really sure. Anyway...I fortunately had a copy of the confirmation, and one quick phone call by the NZ agent straightened everything out. Perhaps all this could have been accomplished without having my printed confirmation, but it sure seemed to make it easier.