There is much less in the way of social security here than in Europe.
Your contract with your employer will vary but there is a statutory requirement that you are given some 'special leave' after 6 months of employment. This can be used for sick pay or other purposes such as funerals agreed with your employer in advance.
You will often need a doctors note for even one day off (which will set you back about $50).
ACC will pay 80% of your income if you are unable to work due to an accident within NZ (or in some cases elsewhere if you are NZ resident). This cuts in after a week.
You can also buy income protection insurance which is very cheap given that for younger people, ACC covers the main reason for not being able to work. Two types of policy exist - the sort which pays you 56% of your income (but is not taxable) and the sort that pays you 80% (but is taxable). Either way if you are on higher tax they work out the same amount of money in your pocket. Ask an advisor when you get here.
It is also wise to consider private medical insurance.
AFAIK You would only be eligible for unemployment benefit after two years residence and as in Europe insurance to protect you from this would be pricey.