Why would a foreign military aircraft be parked at a US Air Force base?
It is actually fairly common for NATO aircraft to be at Eglin AFB.  When I was stationed there (a long time ago) we had two RAF Tornadoes on site for several months.  

Eglin AFB is the home to the munitions development programs. Part of that development includes ensuring the weapons are compatible with the aircraft that will deploy them. Eglin also has a large test range to support that effort.  

Since we sell the majority of our munitions throughout NATO (common weapons reduce the  size of stockpiles and increase flexibility), they also need to be tested on NATO aircraft. The Eurofighter is likely there for testing.