Only a very limited set of the DoD shuts down when Congress doesn’t pass a budget. Efforts related to national security (which most of DoD falls under) continue regardless. A “police officer for the Air Force in Kansas” has little to worry about, even if he’s a contractor. National security functions continue when the government shuts down.
Also past shutdowns didn’t represent a “missed paycheck” for those affected, but rather a delayed one. Everyone got back-pay when past shutdowns ended. This isn’t a guarantee - Congress has to pass it as part of the spending bills - but it has always happened.
Millions of federal civilians and contractors will be furloughed during a shutdown, and that’s a very bad thing. But the military angle in this article is just plain false.