My time to shine, all those hours in books about Rome and Facism will pay off now! The Italian and German Eagles are based on the Roman Aquila Standard. Here is a Nazi Standard, compared to the Roman (replica) one here. Both Hitler and Mussolini were heavily inspired by Roman traditions. This includes "Sieg Hiel" and the Nazi salute, which come from the Roman Salute. The roman salute, however, is not proven factual, but Hitler still used it. (As did we... See the Bellamy salute.) Mussolini's eagle carries a Fasces, another traditionally Roman symbol.

The Aquila is an extremely common symbol throughout Europe. If you take a look at the flags/standards/crests of countries like Albania, Poland, Prussia, Italy, and Germany, you will find some incarnation of the Imperial Eagle. Alot of the newer eagles (Medieval onwards) are actually in fact based on the Standard of the Holy Roman Empire: http://en.wikipedia.org/wiki/Holy_Roman_Empire#mediaviewer/File:Banner_of_the_Holy_Roman_Emperor_(after_1400).svg) Even in Germany today, they still use it, under the name Bundesalder

As for it's use in American culture, I can say that it stems from the fact that the Bald Eagle is often representative of the nation, and our founding fathers were also heavily influenced by the Roman Republic. So, no there is not any connection between all of these except that the Eagle is often seen as a nationalistic symbol, due to its ties with Ancient Rome. Here are Fasces on the symbol for the National Guard: http://en.wikipedia.org/wiki/File:National_Guard_Bureau_(insignia).svg. The Fasces is also found on the Dollar bill, in the clutches of the Eagle, and also on the infamous Mercury Dime, which had on one side, the Roman god of merchants, Mercury, and on the other a Fasces.

This article explains this topic quite well: Jupiter's Legacy: The Symbol of the Eagle and Thunderbolt in Antiquity and Their Appropriation by Revolutionary America and Nazi Germany

EDIT: Mods, I realize this needs expansion, I will get right on it when I have access to a computer.

EDIT 2: Oh god the formatting. I've got to finishing inventory on this last collection and eat dinner, but then I will fix it.