I operate an online business in New Zealand and use BNZ Currency Select for e-commerce merchant services.
Your citizenship isn't the issue. It's whether you can satisfy the bank's account opening criteria. I am not a bank employee and do not speak on behalf of the bank but my understanding is as follows:
2 forms of ID one of which has to be a passport, driver's license or other government issued photo ID
Proof of address
It is a requirement for all banks in NZ (and any developed country) that the ID has to be verified to an acceptable standard. This would normally be done by the bank officer but in your case...? BNZ may accept documents authenticated at NAB (National Australia Bank) if you are across the ditch. You will need to ask them. You may also try at the NZ embassy and pay them a fee.
There is something else I also suggest you investigate and that is the merchant services criteria. I can advise you they will look into your account history (both personal and business) and do credit checks on the company and all directors.
Without a previous banking history with BNZ I really don't know how you will fare TBH.
All in all, unless you have someone on the ground here you will be fighting an uphill battle. AFAIK you can't even (legally) incorporate a company now without at least one local Director.
It is one of those things you may well have to travel to achieve. All the best.