I find it very difficult to justify most historical claims of anticipatory self-defense - it usually looks to me that it's an aggressor using an excuse to justify their aggression. I haven't seen nearly enough evidence to suggest Israel wasn't the aggressor in the Six Day War. While the military mobilization of their neighbors certainly contributed toward Israel's mobilization, that alone isn't justification for invasion. Nasser thought Israel was preparing to invade Syria, but he didn't preemptively invade Israel, he lined up his troops on the Israel-Egypt border and waited. We know now that Israel was not mobilizing troops on Syria's border, but Nasser's choice to defend his border was reasonable and nonviolent, even with false information.

But aside from that, I think it's reasonable to suggest Israel would have attacked even had there been no mobilization of troops from the Arab states. We saw Israel attack Egypt during the Suez Crisis where they forcibly re-opened passage through the Straits of Tiran, their only shipping route to the south other than the also-Egyptian Suez Canal. Just prior to the Six Day War, Egypt cut off Israel from the Straits of Tiran again, something Israel publicly called an act of war. It's not a coincidence Israel went ahead and took Sinai (immediately adjacent to the Straits of Tiran) during this war and didn't give it back until the Camp David Accords. (It's worth noting that had Nasser not gotten the original false information, he wouldn't have done any of this, and it's entirely possible the entire thing would have been averted. But he did, and that was a huge blunder on his part. Still, I disagree with Israel that refusing them passage through shipping routes is an act of war.)

I would also suggest that Israel's behavior after the Six Day War doesn't seem like the actions of a country that was acting in self-defense. They conquered land during that war and continue to occupy most of it to this day. They've invaded other countries since, with stated reasons that are as believable as the United States' reasons for invading Iraq. They've continued to occupy additional land. These actions indicate a country interested in expansionism and power growth, not peaceful co-existence.