Poland, the Habsburgs, Hungary, the Byzantines, Russia, you can’t go anywhere save Bohemia without tripping over an eagle. Bohemia, just to be different, has a two-tailed lion.
The Hungarians claim descent from a steppe princess who was seduced by an eagle and bore five sons. They were the founders of the five major clans that went west and eventually took over the Pannonian Plains. This was recorded in the early Middle Ages, by churchmen, so any influence from Greek mythology… However, given the traditions of totem animals among steppe peoples, I wouldn’t bet against there being some pre-Christian core in the legend.
Others drew on Rome. The eagle is a symbol for kingship and power in many cultures, and Rome adopted it as one of their symbols, putting it on their insignia. The double-headed eagle seems to have come into Europe from Mesopotamia and Anatolia, where the Romans encountered it, then the Byzantines took it up. It could show the East/West joining of the Roman Empire, and Rome’s claims as the successor of older empires (Alexander’s, among others). The double-headed eagle as insignia of the Holy Roman Empire (King of the Germans) appears in the late 1200s.
It got a little fancier with Charles V.
Poland, being a bit more modest, and caught between three empires, settled for a single eagle.
Eagles look back to Rome. They also nod to other mythologies, Slavic, Germanic, to Christian iconography (St. John the Evangelist is depicted as an eagle). Only monarchs could use eagles as their birds when hunting. The legend of Poland’s eagle goes back to the founder Lech and his attempts to catch a white eagle’s offspring.
Why the lion has two tails… sources vary. Moravia kept the eagle, but in a red and white checkerboard pattern. No, not on. The eagle is checkerboard.