I'm not quite sure, but I think it has something to do with the emphasis. I think if you say 'Zie je me' there is emphasis on the 'Zie'. But I think you can also say 'Zie jij mij' then there is emphasis on the persons. So with 'je' you use 'me' I believe. Also it sounds better 'hij mij', 'je me', 'jij mij'... But that is subjective of course :)
Why is "Ziet hij me?" an unacceptable answer?
Thanks for the help :D