That estimate was based on the only successful efforts at ending illegal crossings--Operations Hold The Line and Operation Gatekeeper. The coverage in those instances was one agent per ten yards, or 176 agents per mile of border, actually on post at one time.
BTW, that's not 42,000 servicemen to get 14,000 per shift, unless you intend that those servicemen never get sick, never need to perform other duties such as Barracks Duty NCO or mess duty, never get a weekend off, and are never permitted to take leave time during their enlistment. It's actually 70,000 (it takes five bodies to man one post 24/7, not three). And those 70,000 would be enough to man, at levels known to be effective in stopping illegal immigration, a whopping 56 miles of border.