No, the qualification for being Jewish (and this is the only answer based upon Torah law ), is if you are born of a Jewish mother, you are a Jew, period.  Whether you observe, don't observe, renounce,convert to Catholicism, join a Buddhist monastery (like Leonard Cohen),etc...doesn't matter.  Born of a Jewish mother...Jew.  Jewish for life.  Biological Father Jewish, biological mother Gentile...the child is not a Jew.  Orthodox Judaism does not recognize children born of Jewish fathers (where the mother is not a Jew) to be Jews.  To become Jews, such children must convert (yep) and the conversion must be an Orthodox conversion to have any value.  Orthodox conversion process involves study, religious practice and a ceremonial bath in a Mikva (special bath house where the bath water consists purely of collected rainwater).  

Zionism is a nationalistic/political movement not a religious one...and in that sense their position is understandable.   They are against the State of Israel representing them as Jews (on religious grounds).  Nonetheless the reality of their situation (and I think all Jews) is that without Israel's presence, world Jewry would always be more threatened.

Read this article....it explains the laws about Jewishness.

http://www.beingjewish.com/identity/whoisajew.html

Gene