USA apple store blocks free NZ app

Yes you can. I would suggest that you set up a separate Apple ID to use for your NZ account. However, you may not be able to change if you are using certain services, have an open Apple subscription, or have a balance on your current iTunes account:

Change your Apple ID country or region - Apple Support

You won't lose your US purchases, and anything that is physically on your device will remain there. However, you wouldn't be able to access your "past purchases" (anything in iTunes in the Cloud) for your US ID until you signed back in with that ID.

The other thing you want to keep in mind, however, is that if you were able to set up and sign in with an Apple ID that was tied to the NZ store, and while that account was signed in you did any of the following, it would "Associate" your device to that Apple ID for 90 days, and you would not be able to do any of the things on the list below with your US Apple ID until the 90 days were up:

Downloading Past Purchases

Using Apple Music

Using iTunes Match

Being in a Family Sharing Group

Have Automatic Downloads turned on

If the only thing you do is to purchase a new App, then the device would not become Associated with the NZ ID:

View and remove your associated devices in iTunes - Apple Support

Best,

GB

Unless you have an Apple account with a New Zealand Credit Card and a New Zealand mailing address, you cannot sign into or use the New Zealand iTunes Store.

Best,

GB