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
What is the NZeTA
The NZeTA is a pre-travel authorisation for visa-waiver passports, not a visa. What it covers, who needs one, and what it does not do.
How to apply
Applying takes about ten minutes in the official app (NZ$17) or on the Immigration New Zealand website (NZ$23). Here is the order of the steps and what trips people up.
Cost
The NZeTA request fee is NZ$17 in the app or NZ$23 online, plus the NZ$100 IVL — about NZ$117–NZ$123 in total.
What you need
A valid passport, a photo of your face, an email address and a payment card. What the photo has to look like and what else is asked.
Processing time
Most NZeTA requests are decided in minutes, but Immigration New Zealand asks you to allow up to 72 hours. What makes an application take longer.
Validity
An NZeTA lasts 2 years from the date it is issued, or until your passport expires — whichever comes first — and allows multiple entries.