No but according to the NZ authorities they are going ahead with it because
"The EU is New Zealand’s third-largest trading partner, with two-way trade valued at $19.6 billion in goods and services in the year to June 2015. This comprises over $8 billion in exports ($4.9 billion goods, $3.1 billion services), and $11.5 billion in imports ($8.5 billion goods, $2.9 billion services).
The EU is our second-largest source of foreign direct investment in New Zealand (NZ$10.9 billion) and third-largest destination for New Zealand direct investment abroad (NZ$2.89 billion). Nearly 40% of all New Zealand international research collaboration is with EU partners. New Zealand is now one of only six WTO members without an EU FTA in place or under negotiation. This puts New Zealand at a competitive disadvantage in the EU market. Similarly, the EU is the only one of New Zealand’s top ten trading partners without an FTA in progress." More can be found out here
https://www.mfat.govt.nz/en/trade/free-trade-agreements/agreements-under-negotiation/eu-fta/
They will probably not have to accept free movement as such but it will probably have to accept something like the Canadians are doing to get their free trade agreement. The EU/NZ FTA is expected in 2017 but CETA is being seen as a bit of a template.
However, this is not access to the common market as an EU member this is just a trade agreement and potentially all 27 national parliaments will have to vote to accept it.