I lived next to Eglin for years and dove in the area but I never heard of a cave on the base. Having said that, there are two parts to Eglin Base. The actual base itself and the area around it known as the Reservation. The Reservation is huge and has multiple fresh water springs on it that flow deep, fast, cold and clear. (Great snorkel trips) I never heard of any caves in the area but I imagine there has to be because of the springs.
A little bit North of there up towards Marianna and Chipley, are some deep caves. I was part of a crew surveying one on private property. We found an underwater lake that had some flow in it but we didn't bring diving gear.