First let me be clear, I'm in favour of Covid 19 vaccines. Sure, a tiny minority have severe side effects and a small number of people can't take the Pfizer vaccine (and that's important, because government should not act in ways that are harmful to such individuals), but by and large it is highly beneficial for there to be widespread vaccines to reduce deaths, hospitalisation and illness from Covid 19.
The bigger philosophical and political question is whether they should be compulsory or whether the state and private businesses and citizens have the right to require vaccination status to be demonstrated to access their services or property. This is where individual rights collide, and the role of the state SHOULD be to ensure the proper delineation between those rights.
For any owners of private property it should be very simple, it should be up to the owner as to whether or not to require vaccination status to enter that property, whether as a customer or not. It is your property after all. The basis for your decision should be up to you and the public will decide whether or not to go there. Assuming New Zealand achieves vaccination rates in at least the high 80s%, then many will decide that it isn't safe to enter some premises that are laissez-faire about vaccinations, other may be relaxed about doing so. So be it. It is private property.
What about employment? Any private employer that wants to only hire people who are vaccinated, should feel free to do so, as long as it is explicit in the employment contract that vaccination is required or that the employer may, from time to time, require employees to take preventive steps to protect other employees or the public. This should be clear for any new hires or any existing hires with such a term in their contracts. For existing hires with no such terms, it is problematic to require vaccination, but it is not problematic to take other health and safety measures voluntarily if there is concern about a non-vaccinated employee, or indeed if there is a risk to the business because customers do not want service from a non-vaccinated employee. Ultimately, a business should not be able to force an existing employee to get vaccinated, but that employee also cannot force the business to act in ways that undermine it.
So what about the state sector? If it is treated like the private sector, then the same rules should apply to employment. New employees can be required to be vaccinated and existing ones cannot, unless there is provision in their contract to enable it. However, given the state imposes lockdowns on the entire population and businesses because it treats Covid 19 as a national emergency, it seems only reasonable that those that work on the frontline, for the state, in enforcing this, are required to take steps to minimise transmission of Covid 19.
Those working at the border, Police and other emergency services, managed isolation and quarantine all should be vaccinated, as they are at the frontline of the state's strategy to contain Covid. Beyond that, it is rational for all working in the public health system to also be required to be vaccinated (although it should be possible for private medical professionals to operate, without taxpayer funding, without vaccinations if they so choose). Taxpayer funded private facilities should be little different, except that a private facility should be able to opt out of receiving taxpayer funds if it wants to operate sans-Covid vaccines. That's private property rights.
What about everything else? Teachers and school staff should be a matter for the owner of the schools. The state can mandate vaccines for state schools and require those who want to continue receiving taxpayer funding to have such a mandate, but it should not mandate it for fully independent schools (or anyone providing private tuition).
Why does this matter? Because private property rights, contract law and personal sovereignty matter. You should absolutely be able to prohibit anyone from accessing your property, including business, without being vaccinated (or if you so wish, if they are vaccinated), but you reap the consequences if nobody wants to go there. You should absolutely be able to choose only to hire people who are vaccinated or who are not, but existing employees should not be forced to be vaccinated, unless there is provision in their employment contracts enabling this. Employers might change the duties of the unvaccinated, and take steps to protect other staff or customers if need be, and if the business loses customers because it doesn't have a fully-vaccinated staff, it might also decide if it needs to make staff redundant as a result, but it shouldn't come to compulsion.
Further to that, whilst it is entirely consistent with the defence of a country that entry into it can be made dependent on both Covid tests and proof of vaccination, it should not be necessary for citizens or permanent residents (but other options, such as managed isolation, can be used to protect the country from infection). It should also not be necessary to have a "vaccine passport" within the country's borders, except for businesses that choose to use it to enter their property (that includes airlines and bus companies).
So no, there should be no mandatory vaccines for private citizens not employed by the state, nor mandatory vaccine passports to travel internally, but property owners and individuals have every right to impose their own rules on who they allow onto their property, who they hire, trade with and interact with.
You don't have a right to force someone to get vaccinated, but you also don't have the right to force someone to employ or trade with you if you choose not to.