2 years later and it looks like online checkin is still a fail (never-mind that the app dumps you out to the website to do checkin):
"To issue you with a boarding pass for this flight we are required to view an onward ticket out of New Zealand. Present your onward ticket to an Air NZ representative in the check-in area to complete check in. "

It's a round-trip ticket, the return segment is on the very same ticket. Sigh... (The return is on UA, so at least I'll very probably be able to checkin for that segment online, or in the app.)