The black tank flush is an interesting topic whether it "needs" blown out or winterized. By all means, it can't hurt at all to do so. However, I believe this is a self contained system with only one check valve. After flush water on tank side will drain into tank by way of gravity flow. Water on input side will drain back out input port by same. It's an open system versus closed. I saw a diagram on it and that's how I recall it. Maybe someone on here can confirm for me?
The only possible concern would be if the black flush water inlet has a check valve similar to the main water inlet. If so, the water on that side wouldn't drain.
Also, the vacuum breaker behind the shower is a bit of an unknown. Could it retain enough water to suffer freeze damage? Probably not. But I've often wondered if the leaks on the vacuum breaker might be from freeze damage.
In order to push antifreeze into the black tank flush, you'd need to pump the antifreeze through the outside shower spray head while the shower spray head is connected to the black tank flush inlet. Of course to do that you'd have to have an outside shower with a removable spray head and standard hose fitting.