Ok quick reply and an "address" of the stance and the whys and hows and "what-fores"
I am a former US Marine and I was a close combat instructor for the Marines and later I did the same (and more) instruction including writing a few SOPs for the Department of the Navy. I still teach classes 3 times a month and I have been doing this type of work on and off since the end of the Vietnam war. So what I say here is not just an opinion. I have real knowledge of this subject.
The square stance is for keeping a chest plate (body armor) square to the enemy. Glancing blows of incoming rounds can run off the plate and hit arms (brachial artery) in theory, and it was found in testing that a square impact was actually safer then an angled impact for the soldier.
"Thumb over top/high arm" was developed specifically for the M-16 and M4 weapons, for close quarters fighting using full auto, for room clearing. Having the weapon under some weight (the arm) keeps the muzzle down in very rapid fire at close range on relatively flat ground. Pistol distance. A full auto burst of 5.56 rounds is far deadlier then a hit or 2 with a pistol. It doesn't work well with most other weapons for various reasons, from heat to recoil to exhaust gas and so on.
So.....
If you want to fire very fast at close range with a light caliber and be very deadly in a fight, with an AR or M16 variant, the "high arm/thumb over top" is an excellent way to shoot.
And IF you are armored up and IF you are facing an enemy who is likely to shoot back, and IF you are shooting an AR15, then "squaring up" is also a good way to shoot.
But for good accuracy in all other scenarios of rifle shooting the standard "NRA" positions are still the best.
For a BANG at 99.999% of the targets a civilian will ever shoot at, the basics are best, instead of the need for BANGBANGBANGBANGBANG at an enemy shooting back at you, in a room, while you just happened to be wearing a full battle kit and armor.
I do not say such a technique should not be learned. Not at all. It's a tool in the tool box that you can pull out and use if you know how, and there may be a time when it's useful. The times its used the most for anyone who is not in combat is "Running/gunning competition. I teach it myself, but only in context.
As we see with Jerry Miculek, it can be used at long ranges too, but it had NOTHING to offer at long range that the standard "NRA" positions can't do better. It's a trick he does to demonstrate that is is a useful technique and that it can be learned and used well.
But it's taken on the Fad status today. Emperors new clothes, in all their glory.
Today we see it over and over and over because it makes shooters think they are on the cutting edge of technique. So be it. It's not harmful. That cool-aid tastes good to them.
But to tell the truth the square stance and the High Arm/Thumb Over Top way of shooting is only "best" in about 1/10000% of the shooting you will ever do in the real world.
You won't see it used by David Tubb at any range over 20 yards.
You don't ever see it used at Camp Perry.
It's not used deer and elk hunting.
It's not used against dangerous game in Alaska or Africa.
It's not used in NRA small bore competition.
And it's not used by the US Marine Corps, US Navy SEALS or US Army in combat except when shooting inside rooms and/or at rock throwing distances, and then only when they are armored up.
Don't loose sight of the objective and then redouble your effort.