The material is still classified, so for the military at least it doesn’t matter where they view it. It would be equivalent to finding a “secret” folder on the sidewalk and deciding to take a peek. If you’re military and you don’t have permission to view material intentionally doing so is breaking the rules.
Yes, absolutely. My point, perhaps clumsily made, was that the Army isn't monitoring all Google searches (which would be in violation of posse comitatus), but is monitoring google searches on government networks
As a government worker, you are to avoid these cables no matter where you are, or where you view them. That's (at least by the letter of the law) forbidden.