Physical punishment was used in the American Army officially till 1861. It is still used unnoficially. I can see the AFFS also using it, as it has a machismo tradition.
Uhhh, as a US Infantryman from 1993-1997, and 2001-2005, I can say that it is *NOT* used, even unofficially.
There are also serious differences between "punishing" an enlisted, NCO, and officer.
Officers get bad OER (reviews) which can kill their ability for promotion.
NCO's get bad NCOER's (reviews) which can prevent them from hitting E7 (there are many bad E6's in the army)
Enlisted get bad counseling statements, but it doesn't mean anything. Typically they can get put on "details", which are usually trash/kitchen/guard related, or in garrison for stupid little things they get "smoked", which means lots of push-ups, flutter-kicks, and whatever else a creative sergeant can think of to make the enlisted soldier think twice before doing whatever it was he did again. When I was enlisted, I did a LOT of pushups for being a smart-a$$, a LOT, and it never taught me anything. But I knew my job, and when I got sergeant I got all sorts of awards and such for being a good NCO.
Individual mercenaries who refuse orders would simply get *fired*, just like blackwater/Xi or any other private security firm. Entire units may refuse to do something against their contract, or that is unlawful, but those are issues for courts when it comes to their contracts.
Mercs are people who join up just to fight, and make money, it isn't in their idiom to refuse an order to fight, and thus make money. It is like professional athletes, they don't refuse to play.
You can read up about really undisciplined soldiers by reading up on the Lone Wolves. You show up in the morning, get a day's pay, fight, and decide if you want to come back tomorrow. But once you get paid, you don't refuse to fight, especially IN combat, else you immediately become hostile.
But in today's military soldiers aren't physically disciplined, and if they are for some reason, a quick call to the IG (investigators) means that NCO will soon be enlisted again, if not kicked out on a dishonorable or serving time in leavenworth.
"Blanket parties", or a lot of soldiers putting soap in socks and beating up other crappy soldiers happened once or twice when I was in basic, and all the victims eventually dropped outta basic, they were just dragging the unit down and typically kinda psycho. I never partook, but in some ways it was justified, and probably saved lives in the long run. The drill sergeants would try to stop it from happening if they knew about it, and if they found out the entire platoon (or company) would be getting smoked for the rest of the night if not the rest of the week.