An NZeTA covers ordinary visits — tourism, family, business meetings. Travelling specifically to receive medical treatment is treated separately.
What is required instead
Immigration New Zealand directs travellers coming for medical treatment to apply for a Medical Treatment Visitor Visa. That is a visa application with its own requirements, not something the NZeTA can be used for.
Falling ill while you are already there
This is a different situation. Being treated for something that happens during your visit is not the same as travelling in order to be treated. If you then need to stay longer than your visit allows, that is a question about extending your stay, and you should raise it before your current stay runs out.
New Zealand's public health system does not cover most visitors. Travel insurance that includes medical cover is worth arranging before you go, whatever the purpose of your trip.
Source: Immigration New Zealand — New Zealand Electronic Travel Authority (NZeTA)
More on the NZeTA
NZeTA or visa?
The NZeTA covers short visits from visa-waiver passports. Longer stays, work, study beyond 3 months and medical treatment need a visa instead.
Which one do I need?
Similar names, very different rules. Which scheme applies to which destination, what each one costs, and the mistakes that come from mixing them up.
Arriving in New Zealand
Your NZeTA gets you on the plane. What comes after: eGates, biosecurity, money, driving, phones, health, safety and holidays — from official sources.
eGate at the border
eGates at Auckland, Wellington, Christchurch and Queenstown take ePassports from 59 countries and territories, age 10 and over. The NZTD is still required.
Liquids at airport security
International flights from New Zealand: liquids in carry-on in 100 ml containers, one 1-litre bag. Domestic flights have no liquid limits.
Money, cards and tipping
New Zealand uses the NZ dollar. Prices include 15% GST or must say they don't, tipping is not expected, and cash of NZD $10,000 or more must be declared.