Define your working definitions of Jewish nationalism and white nationalism. Chances are, you're throwing two incongruent things under the same "nationalism" terminology when both things should not be compared to each other.
People often use the term "Jewish nationalism" to describe merely supporting the Jewish State or wanting it to thrive. For Israelis, this is just mere patriotism, which is like nationalism but without the potential negative connotations one can perceive. For others, it's just admiring the State and many aspects about it, and most often not to the detriment of their own. A stronger outspokenness about Israel may come from the fact that other people are actively trying to annihilate the nation, which fuels a need to be more vocal about their support. In addition, nationalism also stands for separating a state out to be its own (like Scottish nationalism); and, while Israel is its own independent nation, there are multitudes of peoples and organizations trying to end it. This kind of nationalistic spirit doesn't end when a nation is formed and, as attempts to destroy Israel have not ceased since it was formed, that has keep up the same spirit we may call nationalism.
Do note that this support of Israel does not inherently have the anti-Arab, Islamophobic elements people often mistakenly ascribe to Zionist tenets. At its central core, Zionism makes no comment about non-Jews and concerns itself simply with that Jews deserve a home. That's not a belief that excludes non-Jews, which is why there are so many Zionists who aren't Jewish. Even excluding those Christian Zionists who just support Israel for theological beliefs, there are Hindu Zionists, atheist Zionists, Buddhist Zionists, and even other Christians whose Zionism isn't rooted in just their Christianity or certain apocryphal beliefs. Zionism, for most people, perfectly allows equality between races (especially since Jews can be of all races) and does not support supremacy of the Jews. It really only states that the Jews deserve a home, and has enough room to recognize all other peoples as equal.
This non-racial, equality narrative is something Bernie can accept. He still has his criticisms of Israel and how it actually functions; but, the ideals of Zionism don't contradict his beliefs of his stances on the thing we call white nationalism. That's a whole different beast.
White nationalism focuses on a homeland for white people only. This separates itself from the ideas labeled as "Jewish nationalism" because it focuses solely on one race and makes sure to expel or marginalize all those who aren't of that race. Israel, on the other hand, has no legal framework to do that to others, even if we replace race (which the Jew are not and can be of any) with ethnic or religious concerns. At the very least, the ability to convert opens up Israel to anyone, if you believe they're only open to Jews. There is no conversion option afforded under white nationalist beliefs. Israel, the "Jewish nationalist" state, also functions on its peoples' homeland whereas white nationalists don't give a sh*t about doing the same in Europe and continuously oppressing or ostracizing native occupants.
Israel better fits Bernie's ideals because it allows for a much more diverse population. It may be the Jewish State, but it's not an exclusively Jewish state and besides, he's fine with self-determination for other groups including the Palestinians. White nationalism is something he despises because that'll be a denial of things he wants to see guaranteed for all people.