Well, most of the nations you listed have a common history and are from the Western hemisphere (particularly Europe and the US given its heritage can be counted as having a very heavy European influence in its founding despite all its changes).

The Greeks used the eagle because it was the symbol of Zeus the King of the Gods, and the Romans whose religion and culture had early influences from Greek traditions and associated the eagle with Jove or Jupiter as well as the wolf which was associated with Romulus and Mars although the eagle became the primary animal. Further, most Mediterranean cultures shared a common Indo-European origin further increasing the parallels between Greek and Roman mythology. Why an eagle would be chosen out of birds to represent a sky-deity might be due to their status as predators, their appearance, and their wide range of distribution. The Eagle was the King of the Sky and of Birds, similar to how lions were the King of Beasts.

Most of the nations you mentioned (US, Prussia, Russia, etc.) based aspects of their symbolism and government on the Roman model and the Holy Roman Empire in particular wanted to be the continuation of the Roman Empire, hence the name and the fact that the eagle was two-headed like two halves (Eastern and Western) of the Roman Empire. The use of an eagle was specifically because it had been used by so many and held a certain power that a new symbol might not, and because Rome was essentially the standard of Empire that most nations held themselves to and is still widely considered so. Throughout Germanic cultures the eagle was then used due to the influence of the Holy Roman Empire.

Also, the Byzantine Empire was really just the Eastern Roman Empire although it developed a distinctly Greek slant to its culture and traditions, Byzantium is mostly just to make things more convenient for historians.

So, the prevalence of the eagle in all likelihood has to do with its association with two very widely admired and emulated ancient civilizations and then with all the other impressive ones that followed them and used the eagle than with many different peoples all deciding that the eagle made a good symbol.