Both are correct gernt... your choice
According the Cambridge Internatioanl Dictionary of English Language, you can use any of these three terms: Argentine, Argentinean or Argentinian both for people and as an adjective.
'Argentine' was the original British term for both people and things. But Americans introduced later the word 'Argentinean' or 'Argentinian' copying the suffix -ean from other countries of the region (Chilean, Brazilean, Ururguayean, etc.). Nowadays, most people in America would use the word 'Argentinian' not only as adjective but also to refer to people.
In Britain, I guess that most people would say 'Argentine' for both things, but there are many who are now using 'Argentinean'.
It is a matter of tastes.