Thanks for the great question!
In order to find out whether a school or program is covered by the GI Bill, all you have to do is go to the Institution Search page on the Veterans Affairs site. For institutions within the US, you can click on the state. If your school is outside the US, you can click the “search by country” button, and on the following page, select your country.
Doing that, I see that Auckland University of Technology is at the top of the list. You can click on its link to find out pertinent details, such as the maximum amount authorized for tuition under the new Post 9/11 GI Bill, and contact information. I don’t see Manakau Institute of Technology on the list.
If your school isn’t listed, the situation isn’t hopeless. The VA authorizes programs at overseas schools on a case-by-case basis. You will need to contact the VA, using the information on the Foreign Program Approval page of the VA site.