I didn't take anything that came out of Jordan or Egypt as anything serious. If a war were to expand it would do so with an unplanned escalation, not by willful consent by Arab states to enter a war they haven't been preparing for.
IF Hezbollah does make good on its promise to open a front in the North, and the Americans do make good on their promise to bombard Hezbollah, that could lead to Hezbollah sister organization in Iraq (particularly) to bombard U.S bases and military installations, particularly in Iraq and Syria, which again in return could lead the Americans not only bombarding the Iraqi PMF, but could lead to military operation to sever the links between Iraq and Lebanon, the M20 highway in Syria.
Blue: America possible move. Green: line of communication between Iraq to Lebanon (sorry for the bad edit)
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If they decide to do this, then the Americans would rely on a few key airbases, most notably (and closest to the epicenter) is Muwaffaq As-Salti AB in Jordan.
9734+VVF, Al-Mafraq, Jordan
Do this and the militias or even Iran directly could bomb those airbases, dragging those countries into a wider regional war, so if Egypt or Jordan enters a war, it will likely be because they're forced to do so, not because they're so confident in their army.