Members of the IAG group have put out a very informative Earthquake Claim Pack for their customers. The insurance companies making up the group are listed below. Click on the name of the company to be taken to the appropriate web page where you can download the pack relevant to your policy.
ASB don't, as of the 2nd of April, have a pack along the lines of the other companies. There is a more general FAQ page here. IAG have a generic pack here.
Although they are almost identical, the NZI pack is a good one for clear explanations and a comprehensive checklist.
If you are insured with a company outside those mentioned above, be aware the way your policy is administered could be significantly different.