Everything I remember learning about the six day war is that there were a number of diplomatic moves back and forth between both sides, culminating in Egypt's mass border stacking of troops and tanks. Before that was Egypt's refusal to let any ship flying the Israeli flag through the Straits Of Tiran. Look, Britain wasn't threatened by Argentina, but Argentina took an action that pissed the British off-a fairly useless plot of South Atlantic islands. So the British took care of it (and all the while the British troops were steaming towards the Falkland's diplomatic efforts were going on) because they weren't going to let it stand. The situation in the Middle East in 1967 was a lot more complex of course, but do you see it at the most basic level? Egypt et al, took actions that pissed the Israelis off, something that they were not going to give into. So Israel took actions because they were not going to let that stand. It is absolutely no different than our colonists saying Don't tread on me.
You are using the word threatened, and maybe that is the wrong choice of word.