Like the OP we did our NZeTA through the app a couple of months ago and got approved. However, a short time later the app started showing network error. I even installed the app on my work iphone and get the same error message immediately. Updated the apps, deleted, cleared the cache/storage, same result. We do have the confirmation emails that we will print and take with us.
I did discover that you can also verify the status on-line at https://nzeta.immigration.govt.nz/check-status You will need the NZeTA reference number and your passport number. (both are on the NZeTA approval email.) My understanding through research is that the NZeTA is electronically associated with your passport and you do not need to have a physical copy or show the electronic version to anyone. (But having the print out does make me feel better about it.) We are flying into Auckland in a few days.