After the Sorghelli War the falcon to the UGI meant they were ever watching for the enemy to rise. The Falcon on Earth was a hunter, and it was known for its eyesight as well as its ability to "see" the future when it came to weather. With that said the Falcon in the UGI's sense meant they were waiting for change, and will rise to destroy the threat.
The Star within the eagle is called the Lone Star, it is risen on a pedastool in remembrance of all the fallen UGI military and civilians. While the background signified the UGI was broken. However the UGI still believed they were able to be rebuilt.