Fort Clinch was incomplete and unarmed at the start of the Civil War, and the caretaker sergeant serving as the federal government's sole representative at the fort didn't put up much of a fight when Florida Militia showed up shortly after Florida's secession in January of 1861. Robert E. Lee ordered the abandonment of Fort Clinch in 1862, however, as he (probably correctly) viewed it as indefensible in its ungunned state.
Rodman Guns likely didn't appear at Fort Clinch until the tail end of the war, and the fort was never brought up to its intended armament of 70 guns.