Note: A permanent resident includes the holder of a permanent visa under section 30(1) of the Migration Act 1958. New Zealand citizens holding a special category visa under section 32 of the Migration Act 1958 also satisfy this requirement.
Note that the use of "permanently residing" does not mean a permanent resident - it means you mainly live here in Australia.A first home buyer who fulfils the criteria below:
* At least one of the applicants must be an Australian citizen or have permanent residency in Australia. New Zealand citizens permanently residing in Australia, who hold Special Category Visas, may also apply.
Each State's lodgement guide that I could find online specifically mentioned that NZers here with an SCV were eligible alongside PRs (as long as you meet all the other criteria, obviously). So whoever has told you you aren't eligible for the FHOG hasn't followed the guidelines (unless there is another criteria on the list that you don't fulfill).
I don't have any experience with the health care card but it sucks you haven't been able to get one. If your daughter is a Kiwi & isn't eligible for help from the Aussie government, is she eligible for some sort of help from NZ? The reason I ask is that happens with the pension - Kiwis get their pension from NZ because they don't get it from Australia.