Esser Agaroth (2¢):
As far as the State of Israel, and its army [IDF] being Jewish, well, let's see....
1. Fighting for democratic (Hellenist) rights, instead of fighting for us to perform the Will of The Almighty.
2. Continuing to insist on being just like the goyim (non-Jewish), a member of the international "family of nations," instead of striving to be the separate and distinct people we are supposed to be.
3. Watering down Shabbath, kashruth, marriage and conversion regulations, for the sake of democracy.
4. The indoctrination of even religious children into Israeli, secular society, to make sure that the authority of the State is central, instead of the authority of The Almighty through His Holy Torah.
5. The suicidal giving away land, not even ours to give away, and to hostile enemies no less.
6. Increasing levels of romiscuity in the IDF, due to male and female close contact in fighter units.
7. Sacrificing the lives of our IDF soldiers for the sake of protecting enemy "civilians" instead of following halakhah related to war.
8. Feminism, egalitarianism, and other "do what we want" libertine concepts.
9. The silence of the State-employed "rabbis" on most of these issues, acting little more than puppets, rather than the halakhic authorities they are supposed to be, how on earth can you call the State and its army Jewish?
Is this what you mean by "Jewish?".
Who was it who said that we'll have a truly Jewish state, when we have Jewish street cleaners, Jewish Prostitutes, and Jewish criminals?
What I mean by "Jewish" is that of being Torah-focused, which is the primary purpose of the Jewish People.
As a religious Jew, I would not want my son entering such a an un-Jewish army, unless it was for one of these reasons, such as having influence on other soldiers for the good of the Jewish People and its eternal Homeland.
The State is only occasionally Torah-focused, and then only cosmetically so. Thus, when it comes to being Jewish, the State of Israel is only occasionally Jewish, and then only cosmetically so.