How is the eagle and its symbolism significant in Judaism?

I am assuming you are referring to the animal known in Biblical Hebrew as the Nesher. I say this as there is good evidence that in fact the proper identity of this animal is the griffon vulture.

Regardless, the Nesher is indeed an important symbol in Judaism. Most significant is perhaps that G-d describes the manner in which He saved the Israelites from Egypt as “....on the wings of the Nesher...” The Talmud identifies the Nesher as the highest flying of all the birds. In addition, the Nesher carries its young on its back (as opposed to other birds which carry in their claws). This is seen as an allegory for self-sacrifice for its children, in that any arrows shot at them will be absorbed by the parent while the young are safe on its back. So too, G-d protected the Israelites from their persuers and other dangers of the wilderness.

In addition, there is a piece in Pirkei Avos (Ethics of our Fathers) (5:23) where in Rabbi Yehuda ben Teima exhorts Jews to be “Kal K’Nesher“, light as an eagle, in the service of G-d. One explanation (Tur) is that just as a Nesher soars effortlessly through the sky, changing direction at will, so too should we soar through life easily turning aside from enticements towards sin.

R’ Daniel Fleksher