The Kiwisaver scheme has provision for contributions to be halted (a contribution holiday) or funds withdrawn in cases of significant financial hardship.
If you need to have a contribution holiday (i.e. take a break from contributing to the scheme) IRD can arrange this for you. The information is on their Kiwisaver web site here.
I am not a financial advisor and claim no particular knowledge or experience with Kiwisaver. Please treat the following as informal information that you will have to investigate for yourself. The circumstances under which you can withdraw from Kiwisaver depends upon the scheme you have entered into. As a generalisation the scheme will have a provision for members to withdraw in the event of significant financial hardship. Surviving a catastrophic earthquake, losing your job, running out of money for temporary accommodation, family experiencing trauma ought to qualify. Work is being done to make withdrawing, because of earthquake related hardship, simpler, and quicker (see this NZ Herald article here).
IRD, at their Kiwisaver web site, have information on significant financial hardship which is here.