Which island country is the only country in the world with all the different climatic zones?

The country is New Zealand.

New Zealand's climate is complex and varies from warm subtropical in the far north to cool temperate climates in the far south, with severe alpine conditions in the mountainous areas.

Interestingly, Mean annual temperatures range from 10°C in the south to 16°C in the north of New Zealand. The coldest month is usually July and the warmest month is usually January or February. In New Zealand generally there are relatively small variations between summer and winter temperatures, although inland and to the east of the ranges the variation is greater (up to 14°C). Temperatures also drop about 0.7°C for every 100 m of altitude.

http://www.niwa.cri.nz/edu/resources/climate/overview

As far as I can see - from this map, at least - New Zealand is entirley within one climate zone.
http://www.mapsofworld.com/world-maps/image/climate-world.jpg

New Zealand is entirely within the temperate zone.
https://en.wikipedia.org/wiki/Image:World_map_temperate.svg

No island country is situated within all 4 different climatic zones.
Australia, Taiwan, Greenland and Madagascar are situated within 2 different zones.
https://en.wikipedia.org/wiki/Temperate

Also when using the more sophisticated Köppen classification of climatic zones it is hard to see that New Zealand is situated within more than one zone.
https://en.wikipedia.org/wiki/K%C3%B6ppen_climate_classification

This answer doesn't depend on latitude, but on altitude, proximity to large bodies of water, and the presence of mountain ranges, plains, or other geographical features.

New Zealand is probably correct if the question is valid, by which I mean that several Central or South American countries which are not islands may have all the different climates. Tanzania is also a candidate. "All" is hard to pin down, by the way. Many countries have a wide range of climates due to geographical features other than location.

https://en.wikipedia.org/wiki/Climate_classification

Good point, but it is still hard to see that there are any island countries within ALL climatic zones, for example as categorized by Köppen.

Here are the Zones:

* 2.1 Köppen classification
o 2.1.1 Tropical rain forest
o 2.1.2 Tropical monsoon
o 2.1.3 Tropical savanna
o 2.1.4 Humid subtropical
o 2.1.5 Humid continental
o 2.1.6 Oceanic climate
o 2.1.7 Mediterranean climate
o 2.1.8 Continental steppe
o 2.1.9 Subarctic
o 2.1.10 Tundra
o 2.1.11 Polar ice cap
o 2.1.12 Desert

https://en.wikipedia.org/wiki/Climatic_zone

If this link works, you will see that Australia is within 6 out of the 12 zones, as categorized by Köppen. Australia is the only candidate among island nations to have such a wide variety of climates.

https://en.wikipedia.org/wiki/Image:Australia-climate-map_MJC01.png