Chittim - the countries lying on the Mediterranean, particularly Greece and Italy (Dan 11:29, Dan 11:30). The Assyrians were themselves to be overthrown - first, by the Greeks under Alexander the Great and his successors; secondly, by the Romans.
Eber - the posterity of the Hebrews (Gen 10:24).
he also shall perish - that is, the conqueror of Asher and Eber, namely, the Greek and Roman empires.