I don't think New Zealand has free internet.
Esquires coffeeshops provide free wi-fi with purchases.
I'm sure Auckland public libraries have free internet because Christchurch ones do
Auckland City Libraries do provide free internet at selected terminals, however unlike Christchurch there is no free wifi connection (you can pay for wifi - which is provided as a Telecom hotspot - but VERY expensive)
I just spent three weeks travelling around NZ and I didn't once come across free wifi. Very frustrating.
Auckland libraries now have free internet and wifi up to 100 mb a day
You can get free WiFi service in about 80% of McDonald's around the country. Esquires Coffee will give you a one hour WiFi access code when you buy anything.