Nobody wants to think about bad things that can happen, but you should ideally always be prepared for the worst.
Good news: Accidents are covered by the state
New Zealand is known to provide excellent healthcare, but its health system works a bit different to most European countries. Any sorts of accidents are always covered by the Accident Compensation Corporation (ACC), no matter if you are a New Zealander or a tourist. If you smash your finger in a car door you just go to any local doctor (called GP, General Practitioner) and describe the accident. They will do the paperwork for the ACC and you most likely don’t have to pay anything.
Things get a bit more complicated with non-accident issues. You’ll have to pay a doctor visit fee, but that’s most likely an insignificant amount. If it turns out that you need expensive medical examinations or treatment, you’ll have to pay that all by yourself though. There might be a cross-country agreement between your home country and New Zealand in place that allows you to get the paid amount (or parts of it) refunded at home, but that can be a long process.
We therefore recommend to get a travel insurance. That doesn’t cost a vast amount of money and keeps things simple when bad luck hits. Those insurances also cover transportation costs back to your home country if that is required.
Southern Cross is always a safe choice as it is one of the bigger and best known insurers in New Zealand, but there are more providers available. Do some Google research. A Southern Cross Travel Insurance for a family of 2 adults and 2 children for 12 months e.g. only costs ~NZ$1,000.
Booking an insurance online is easy. You just fill a form, enter your credit card details to pay and that’s it. Insurance claims are usually handled in a quick and easy way. You tell their agents on the phone what happened and a few days later you receive a TXT/SMS that tells you whether your claim was approved or declined.
Pay your flight with a credit card
Most credit card providers (Visa, MasterCard, American Express) have travel insurance plans included. But check the details. They are mostly very limited and focus on getting you back home in case of a serious health problem. Though, they often include other events that a classic health insurance probably doesn’t cover, such as lost baggage or cancelled flights.