The situation is more complicated than the article suggests. By early June Nasser had responded to increasing Israeli-Syrian tensions by closing the Straits of Tiran to Israeli shipping, removing the UN observer force from the Israeli-Egyptian border, and sending major troop reinforcements into the Sinai. It appears that he wanted to maneuver the Israelis into attacking first, and knew that (a) Israel considered the closure of the straits a casus belli (b) the IDF could not keep its reserve forces mobilized indefinitely. What he did not count on was having the war start with the destruction of the Egyptian air force.
When Jordan began shelling Israeli territory on the morning of June 5 the Israelis sent messages via the UN asking the Jordanians to stop. Because King Hussein believed that he would be overthrown if he remained neutral in an Israeli-Egyptian conflict, the shelling continued and the war spread to Jerusalem and the West Bank.
As for Netanyahu not mentioning the Israeli nuclear arsenal (which probably contains fewer than 100 warheads), it's been US-Israeli policy since 1969 to maintain ambiguity/"opacity" about the Israeli nuclear capability. I doubt Obama or Kerry really want to see that policy jettisoned at this particular moment.