Does the Greek word for Palaces, Megara, come from the Aramean word Magharat or Zagharat “caves”?

Maybe.

There is a plural megara word in ancient Greek, which means “a kind of crypt into which live pigs were thrown during the Thesmophoria festival”. This is related by both Chantraine and Frisk to Hebrew me‘ārā “cavern”, meaning it is Semitic (in all likelihood), and thus related to Arabic Magharat.

The singular megaron “hall” is less definite; it may be related to the town of Megara; it may be related to megas “big”; it may be borrowed from an external language (Frisk rejects the proposed Indo-European derivation by Brugman), and Chantraine notes that a Condoléon thought it was indeed the same word as the plural megara “crypts”. But that’s just one authority relating the two words.

Zagori OTOH is pretty obviously derived from Slavonic Zagore, “beyond the mountain”.