ASBBNZLanternNZIPSISStateThe packs take you through the basics from lodging a claim with EQC, listing the damaged contents of your house, and explaining which bits are EQC's responsibility and which bits are the responsibility of the insurance company.
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 packhere.
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.