Most of the civilians employed by the US military are in maintenance and support roles.
This has increased dramatically in recent years, largely because the weapons systems have become very complex. Training recruits to maintain that equipment takes a lot of time (and money). It has proven to be far more cost effective to hire retired maintenance people who have completed their enlistment and have already had years of training, or to hire civilians already trained by the weapons system provider in the case of a new weapons system like the F35.
Also true of setting up the information systems that the US military relies on. Those systems tend to change rapidly, and it's far simpler to hire trained civilians than to train enlisted soldiers who will leave in a few years.
For the past several decades, civilian contractors have staffed the cafeterias and commissaries on military bases. Those roles don't involve direct military training, so there is no point in losing a trained soldier to fill either of those capacities. A good deal of logistics support is handled by civilians for the same reason... don't need to go through boot camp to learn how to unload a truck or maintain a warehouse.
Combat soldiers such as those provided by the now defunct Blackwater are typically employed not by the US military but the US State Department for personnel and embassy protection. They are also hired as guards by companies working for the US to provide services both to the US military or other entities.
For example, two of the four people who died in the Benghazi attack were former US Special Forces soldiers employed by the State Dept to guard the ambassador. The same holds true for the people hung from bridges in Iraq early in that conflict - they were ex Special Forces people employed by private companies as guards, those companies providing infrastructure repair.
As for clandestine combat operations... in years past, the CIA ran clandestine operations and employed mercenaries to carry those out. The U2 flights over the Soviet Union and China in the late 1950's were flown by civilians. Later SR71 flights were flown by USAF pilots, but those flights didn't typically penetrate another nation's airspace, so plausible deniability wasn't needed.
Whether or not the US military employs civilians for clandestine combat roles today is not at all clear. Bear in mind that such activities fall under the command of the Joint Special Operations Command, who typically do not make their operations public. Special operations require a high level of intense training. The soldiers who retire from special ops: SAS, GSG9, SEAL, US Army Delta Force typically do so because the rigors of their duty has taken a high toll on their bodies, and they can't keep up any more, so they retire and take on the less rigorous jobs of protecting diplomats or private company operations. No longer the best choice for a difficult combat operation.
And keep in mind the consequences to a mercenary. By the rules of war that most major nations adhere to, any soldier captured out of uniform, or anyone captured who can't prove that they are a member of a recognized military (as in no dogtags) are subject to execution as spies. One example of that was the Germans dressed in US MP uniforms who went behind US lines during the Battle of the Bulge to create confusion. Around 80 of them were captured, all were shot.