Does the sentence "X is New Zealand owned and operated." require any hyphenation?
No - for the following reasons:
Nouns and adjectives are sometimes linked by hyphens to avoid ambiguity, but that is normally done (i) when the adjectives precede the noun to be qualified; and (ii) when the absence of such hyphenation may result in ambiguity and/or difficulty in reading or understanding.
Is the sentence ambiguous without the inclusion of hyphens? No! So nothing would be gained by adding them.
There is an added difficulty because "New Zealand" is a two-word name. Hence any hyphenation of "Zealand" with the following word (as in the Question's second & fourth suggestions: "New Zealand-owned and operated." and "New Zealand-owned-and-operated.") necessarily imply (i) that the word "New" is separate from the word "Zealand"; and (ii) that the word "New" qualifies "Zealand-owned" or "Zealand-owned-and-operated". Consequently, if the words were to be hyphenated in either format as proposed in the Question, then "New-Zealand-..." would also need to be hyphenated on the same basis.
Additionally, the second suggested format is inappropriate because it ties "New Zealand" to "owned", but not to "operated".
If I went for any hyphenation, it would be for the third version, "New Zealand owned-and-operated", because "New Zealand" is there clearly being used an an adjective to "owned-and-operated", and the sentence is clearly seeking to convey that the company is "New Zealand owned and New Zealand operated".
But it is unnecessary, inelegant, and adds nothing useful.
In conclusion, I would suggest that the Question seems to assume that there ought to be some hyphenation - perhaps because there are compound adjectives, and then wonders where to put it. I agree that, in some cases, it maybe semi-automatic and/or appropriate to hyphenate compound adjectives, especially when they precede the noun. In other cases, they might be included for clarity.
But here, there is no ambiguity and no need for hyphenation, so why include any - especially as there is no clear hyphenation format that is appropriate.