(probably a wrong answer; see FumbleFinger's answer)
No, you can't, because you have two nouns on the both sides of "is", and because "New Zealander" is a single countable noun.
If you had an adjective, you would have used no article:
Michael is tall.
If the word "New Zealander" had been in the plural form, you would have also used no article:
Michael's classmates are New Zealanders.
Some nationality words, like "Russian", can be used either as a noun or as an adjective:
She is Russian. (adjective)
She is a Russian. (noun)
They are Russian. (adjective)
They are Russians. (noun)