This question is being asked out of curiosity
The hard truth is you're most likely going to have a hard next few years. Your best bet may be work at a hotel/restaurant at the entry level, one where you don't have to fill out an online application, put together a strong body of work, finish school, over the next few years to where you are qualified for management, at which point your discharge conditions matter less. The hospitality industry is usually more forgiving of peoples' pasts.
Not sure why you would even ask this except to avoid something. Employers do obviously research prospective candidates and the predominance of information sites would obviously reveal. Remember honesty is the best policy.
The short answer is yes, "they will know". Any form of background investigation, even something as simple and quite common as a credit check would likely uncover your ties to the military. Also, it is not clear how the obvious time gap is addressed with such an omission.
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