I am basing this off of a tape I heard from R' Dovid Orlofsky - he did not cite his source, but a close student of his told me that a lot of what he says is from HaRav Moshe Shapiro.
If we would say "thank you for making me a Jew" in the positive, it would put a certain focus on us as filling that role, as if we were living up to everything that that demanded of us. Similarly with being free and being male, we would be highlighting our occupation of that role and creating an accusation against ourselves if we do not live up to the responsibility demanded by that role. However, we do want to acknowledge that we are thankful to have been given the opportunity to have more responsibilities. Therefore, we say it as a negation of the opposite, in order not to make a direct declaration that "I am a free person."
This is also why women make a different blessing, as in other answers given here - we are thanking for the higher responsibility that we have.