Disclaimer: Some of these views are my own, they may not be accurate, they may be downright wrong in fact – they are opinions. Please research accordingly. This article is not meant to be a historical thesis of any sort, just observations and personal inferences.
With the disclaimer out of the way, I can go crazy with fiction or non-fiction as I please, whee.
I got into a discussion about ancient history today with a senior teacher of mine. It reminded me of how much I used to like stories, myths, legends, and associated symbolism. Anyway one link led to another, and I began reading about Prambanan – a temple in Yogyakarta, Java, Indonesia built around the 9th or 10th century. One of the depicted major gods in Prambanan is Vishnu, one of the 3 major Hindu Gods, whose
consort/vehicle is the Garuda[4.1] – a mythical powerful bird.
In Indonesia, the Garuda is used as a national symbol, a symbol for the Pancasila – the 5 unifying cross-cultural, cross-religious, tenets of Indonesia.
A part of article  and article  tell of Garuda as being the son of a mighty sage Kasyapa and his wife Vinata. Paraphrasing , in short, Garuda had a brother, Aruna, who was born misshapen. Aruna was forced out of an egg (like a real egg, think chickens, not wombs) by Vinata’s (his mother) impatience and overexcitement.
Aruna was angry that he suffered this and cursed his mother. The details of this curse are unclear –  says that the mother became a slave to Naga, I’ll let you read up on it. Essentially, the fix to this situation was for Garuda to steal some heavenly ambrosia – a non-trivial feat (we’ll need to trust the myths on this) – guarded by various beings. Garuda was able to do this and save his mother. This virtuous quality of being able to save one from a disaster underlies (among several other things I suspect) the choice of using Garuda as a national symbol.
The Greeks (and Romans) revered the Phoenix[7.1] “as a symbol of rebirth, immortality, and renewal”[7.1]. USA uses the Bald Eagle[8.1] as their national symbol. According to [8.1], “The founders of the United States were fond of comparing their new republic with the Roman Republic, in which eagle imagery was prominent”. I suspect that the eagle in turn was inspired by the phoenix in those circles. A phoenix is also most closely associated as being the European mythological counterpart to the Eastern Garuda[7.2].
Furthermore, [8.2] speaks of the role of eagles in Native American culture, an ancient culture with their own series of mythology and belief systems. It is highly likely this also factored into the decision to use the Bald Eagle as a US national symbol.
It’s fascinating (to me anyway) that so many places come to similar symbolism, in spite of the vast diversity, and physical proximities between the lands.
Something to munch on on a Saturday. Speaking of munching, I am starved. Exeunt.
 http://my.opera.com/akbar_taksisman/blog/2009/03/30/candi-prambanan (in Bahasa Indonesia unfortunately)
 http://en.wikipedia.org/wiki/Trimurti – Vishnu the Preserver, Shiva the Destroyer, Brahma the Creator. Hinduism is considered to be a polytheistic religion, however, I believe this to be a superficial classification. In fact, many of the older Yoga-based philosophies are very monotheistic. The various divinities and deities being almost individual aspects of the One. In that, it is both monotheistic and polytheistic – the many from the One, and eventually back to the One. It is my personal view, that many of the “modern” groups have, perhaps accidentally, muddled the core essence.
 http://tudtu.tripod.com/garuda.htm – use of Garuda in Thailand.