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 Russian**s**.  (noun)