This site is for discussing established philosophical ideas and texts, what you are asking for is basically an original essay - because how can you expect an answer to exhaust all the points made in a work of art? That's a subjective judgement. See PSE guide on How To Ask A Good Question. Likely this is why your answer has already recieved a 'close' vote.
The resonances with Nietzsche's work have been widely noted. Here are some example essays grabbed at random:
Authenticity is a big theme. Ed Norton's charachter relates to the Last Man, and fear of decay of Western civilisation into mediocrity, and lifelessness. People particularly relate Fight Club's themes to On The Geneology of Morals. It's a big topic. I don't think Fight Club does 'counter' Nietzsche. You could say it explores some of his ideas in a modern context. I don't think it contradicts any of them, or even really critiques them. Durden is a fairly visceral image of an ubemensch, impulsively reforming the world towards his own values. Basically a terrorist. But I don't think Nietzsche would have dissapproved, quite the opposite.
Read some essays, and try to ask something more specific.