How come Syria and Jordan were so easily fooled by Egypt into thinking that they were defeating Israel early on in the Six Day War? Did Syria and Jordan not have their own intelligence or way to find out what was really happening?

It is indeed true that a squadron was used by Egypt's general, Amer, as proof that Israel was losing the war, and a swift counterattack was supposedly underway. At the same time the cable came through, the Jordanian powerful radar station in 'Ajlun was reporting jets as well, making those movements, and Jordan assumed these were the jets Amer was referring to, making their runs to bomb Israel.

At the same time, the Iraqis were saying their jets were bombing Israel, falsely. Nasser himself called Jordan's King, telling him further about the Israeli losses, claiming that Jordan should rush in, otherwise it would be left behind and unable to claim as much territory before the UN intervened for a ceasefire.

Of course, in light of all this, there were other considerations. For one, King Hussein was under significant pressure. He had signed a joint defense treaty with Egypt on May 30, one of the key considerations for Israel in launching its preemptive strike (it viewed this as the final straw proving an Arab alliance was about to attack it). He ran a country that was full of a significant Palestinian population agitating for war against Israel, and had been struggling to balance guerrilla operations by Palestinian groups using Jordan as a base with his desire to avoid war with Israel. If he had sat out a war, it would have likely put his entire ability to rule in jeopardy, especially if he had been asked to join the war. In 1973, the situation was slightly different, in that no request came through, and he'd also lost the West Bank, cutting into the size of the Palestinian population, as well as cracked down heavily in 1970 during Black September and shunted the PLO and its affiliated groups out of Jordan and into Lebanon. These factors all came after 1967, and in 1967, they were all relevant considerations for a king who had seen assassinations of his grandfather by a Palestinian in 1951 (likely due to peace talks with Israel, among other reasons, and the assassin nearly killed Hussein as well) and the deposing of a more distant Hashemite relative in Iraq (Faisal II, executed and overthrown in 1958). So while Jordan may have believed, and had little reason necessarily to doubt, the reports by Egypt in the early stages of the war, it also would've struggled quite significantly to remain outside of the war. Being fooled wasn't the only reason Jordan entered the war, if it was even fooled; perhaps, though it seems unlikely, some evidence exists that they were not fooled but chose to enter the war anyways. We can't know for sure, but given how it reacted to the Israeli counterattack to its shelling of Jerusalem, it seems they truly were fooled by the repeated statements.

One last thing may have served to help fool Jordan. Setting aside the radio reports, the calls and cables from Nasser and Amer, and the radar that seemed to line up with what they were saying, the Israelis themselves were reporting Egyptian attacks over the radio. This was part of a carefully crafted move; Israel wanted to create confusion over the first days of the war to cover its preemptive strike, and allow it some extra diplomatic room to act. It figured that if it claimed it was attacked, this would cause the UN to be more cautious in pressuring Israel for a ceasefire, and it didn't want its hands tied before it could achieve a significant military victory, one it viewed as a way to deter any future attack by the Arab forces. A long, drawn-out war would've been significantly harder to win, so they wanted no one pressing them back and making it harder for them to finish the destruction of Arab forces encircling Israel. This confusion, and the radio reports, may have fed into the Jordanian thinking: when both Israel and Egypt were claiming that Egypt was attacking Israel, perhaps it truly was, Jordan likely figured.

Of course, Jordan's surprise became clear almost immediately, as mentioned above. Israel launched strikes against the Jordanian air force almost immediately after the shelling began. The Israelis had considered a preemptive strike on the Jordanian Air Force, to ensure they couldn't join the war, but the IDF Chief of Staff (Rabin) refused. But when 16 Jordanian Hawkers started attacking Israeli cities, Israel had cause to strike. While the Hawkers were being refueled, Israel destroyed the air bases in Mafraq and Amman, destroyed their runways, destroyed their control towers. This was less than 2 hours after the Jordanians had been running sorties. 40 minutes later, Israel destroyed all of the Jordanian Hawkers (about 20), stuck on the ground. Other military planes were also destroyed, and Israel lost a single plane in the strikes. Hussein himself saw some of the attacks in the distance, and watched a friend of his trying to take off in his Hawker get killed in the airstrikes. Hussein claimed later that he only survived because he was at home; he claimed the palace was attacked as well and he'd have died, as his office was hit. One Jordanian advisor, who had opposed allying with Egypt and signing that treaty in the lead-up to the crisis, said "We’ve lost everything our Majesty built over the entire course of his rule!", and berated the head of the PLO, which was largely an Egyptian-allied group at this time, asking him "And where is the Egyptian air force? Where are your MiG’s, your missiles?"

Jordan was not any better off, in that respect, than Syria. Jordan's shelling of Jerusalem and its attacks on Netanya led Syria to start its own operations, and its planes were striking Northern Israel by noon. They figured that between them, the Jordanians, the Iraqis (also trying to begin operations by then), and the Egyptians, coupled with all the reports coming out of Egypt of great success, that it was time to fight. They too had already allied with Egypt for events like this, and believed they were on the verge of a great victory. Their planes began their strafing of Israeli positions just 50 minutes before Israel's response came, in the form of a complete and utter destruction of half of Syria's air force (over 50 planes), and 10 aircraft destroyed in Iraq's case. Syria was even easier to drag into the war than Jordan; they were far more eager to attack, felt they had a strong position and a smaller border that was more defensible, and they were always more aggressive against Israel. Additionally, it was Syria who was worried about Israel from the start of the crisis; it was they who were told by the Soviets (falsely) that Israel was amassing troops on their borders and preparing an attack. These factors, plus the radio reports and suggestions of success out of Egypt, plus the war frenzy that had already been whipped up in Syria in the weeks leading up to the war, made it quite easy for them to make the decision to start strikes on Israel.