Should be alright. THis is what the US embassy in NZ says:

New Zealand's relationship with the United States in the post-World War II period was closely associated with the Australian, New Zealand, United States (ANZUS) security treaty of 1951, under which signatories agreed to consult in case of an attack in the Pacific and to "act to meet the common danger." During the postwar period, access to New Zealand ports by U.S. vessels contributed to the flexibility and effectiveness of U.S. naval forces in the Pacific.

Growing concern about nuclear testing in the South Pacific and arms control issues contributed to the 1984 election of a Labour government committed to barring nuclear-armed and nuclear-powered warships from New Zealand ports. The government's anti-nuclear policy proved incompatible with long-standing, worldwide U.S. policy of neither confirming nor denying the presence or absence of nuclear weapons onboard U.S. vessels.

Implementation of New Zealand's policy effectively prevented practical alliance cooperation under ANZUS, and after extensive efforts to resolve the issue proved unsuccessful, in August 1986 the United States suspended its ANZUS security obligations to New Zealand. Even after President George H.W. Bush's 1991 announcement that U.S. surface ships do not normally carry nuclear weapons, New Zealand's legislation prohibiting visits of nuclear-powered ships continues to preclude a bilateral security alliance with the U.S. The legislation enjoys broad public and political support in New Zealand. The United States would welcome New Zealand's reassessment of its legislation to permit that country's return to full ANZUS cooperation.

Despite suspension of U.S. security obligations, the New Zealand Government has reaffirmed the importance it attaches to continued close political, economic, and social ties with the United States and Australia. New Zealand is an active member of the global coalition in the War against Terrorism, and deployed SAS troops to Afghanistan, and naval and air assets to the Persian Gulf.