What should I call a person from New Zealand?

You have to use one New Zealander, two New Zealanders... The name is unique in that it is composed of two words. Other countries whose names finish with "-land" don't have two words, e.g.:

Finland, Ireland, Iceland, The Netherlands (The is a definite article), Poland, Swaziland, Switzerland and Thailand, etc.

Among them, only Iceland has its adjective (language) and people with suffix "ic" and "er" as in Icelandic and Icelander. Other countries have Finnish, Irish, Dutch, Polish, Swazi, Swiss and Thai for their people respectively.

The adjective New Zealandic is sometimes used, but not very broadly.

There is no hard-and-fast rule to apply to all people coming from countries ending with -land.

"New Zealander" is often used as a noun, and "New Zealand" attached to nouns where an adjective is needed (e.g. "New Zealand native".

There are too many counter-examples to come up with a general rule. "Englander" is only used in phrases such as: "little Englander" which has a rather specific meaning (an ignorant English nationalist); "New Englander" - a person from New England. In fact "New Zealander" is rather rare in how it forms, probably because there's no nice adjective unlike "English", "Scottish" etc. ("Kiwi" being informal akin to "Aussie" in your original example)