Cases. "mężczyzną" is Instrumental (Jestem mężczyzną = I am a man), "mężczyznę" is Accusative (Widzę mężczyznę = I see a man).
Which version? The English and Polish Wiktionary look fine. Genitive is "mężczyzny".
Why does "mężczyzna" take the feminine -ę ending in the accusative? Is it because it "looks" better?
Well, it doesn't look 'better', it's the only correct form ;) Well, the word is masculine, but it 'looks feminine' and has a declension of a feminine noun. All masculine nouns ending with -a (there is a bunch of them) do.