Me rocking out with my band, which you can hear at www.raindriver.com
No, auto dig in does not apply to non player units. The rule is on page 10 under Non Player Unit Card Deck. Once a non player unit loses its dug in status it will never gain dug in again even with the auto dig in ability.
Even if the SS units could auto-dig-in, they would lose it immediately. The ability is used at the start of Operations, but the SS either move forward or attack forward, either of which removes the Dig-In marker, so it's there then gone right away. The exception is if it's in the US's home row, in which case the game is over anyway.