Wow, some very interesting opinions here. With all due respect, I beg to differ. Zionism is not hard to define, and it is not a religious belief. Zionism is Jewish nationalism, plain and simple. Zionism is a belief that Jews, like all other peoples, have the right of self-determination in their historic homeland, Eretz Yisrael. It is a belief that Jews have the right to govern themselves in their own country.
Defining what it means to be Jewish, on the other hand, is a bit tricky. On the one hand, there is a simple halachic definition: a Jew is a person whose mother is a Jew, or a person who converted to Judaism.
On the other hand, who the Jews are as a group somewhat depends on who you ask. Most Americans will tell you that Jews are a religious group, like Catholics or Muslims. But most Russians will tell you that Jews are an ethnic group, like the Russians, the French, or the Japanese. Surprisingly, while it is perfectly normal for somebody to be both Jewish and American at the same time, the notions of being Russian and being Jewish are mutually exclusive. There are historic reasons for that, which deserve a whole separate discussion.
Personally, I believe that Jews are an ethnic group, and that one can not stop being Jewish any more than one can stop being Italian. While the food, the customs, and the languages of different Jewish communities vary widely, they have always been distinct from those of non-Jews. Because of that, I think it makes sense to talk about a very diverse, but nevertheless a very distinct Jewish culture.
Having said that, I am not dismissing religion at all. The ethnic and the religious components of the Jewish identity are virtually inseparable. It is Judaism that is the common heritage of all Jewish communities all over the world, which provides the core basis for the Jewish culture.
Getting back to your question. To me it makes no sense for somebody who is "demographically" Jewish to say "I am not Jewish...". On the other hand, it is certainly possible for one to be Jewish and to be against the idea of a Jewish state, as much as I would disagree with such a person. Or one can be a non-Jew and agree that Jews, like all other peoples have the right of self-determination, and thus be a Zionist.