When I break boards using my right knife-hand or elbow "from outside to inside" (using a counter-clockwise motion when seen from above), I've always put my right foot forward, which allows for more shoulder & hip rotation thru the target. In the last several years at my dojang, I've noticed multiple people with their right foot back, and several of the lower ranks I've advised to switch their stance have said "so and so upper belt told me to do it this way". Is there some advantage to having the "opposite" foot forward that I'm not thinking of?
A lot of the power in a board break comes from the rotation/torque of the hips and legs to drive through the target. When you start with the same side forward, you have already eliminated most (if not all) of the hip rotation, unless you artificially rotate opposite your stance to then drive through.
When you switch to the opposite foot forward, this gives you the opportunity to step slightly out and past the target, and drive through. Most of the time your impact point will be when the hips are only partway through their arc, which gives you opportunity to drive through past contact point.
It is just a much more powerful strike opportunity when you go opposite foot versus same foot, just because of hip and body alignment.
Example as requested:
Say I am breaking with a straight rear punch, right hand. I want my shoulder in line with my target, and I will step slightly past and out to the left side with my lead left leg. That way my arm is not not fully extended at the contact point, and by stepping slightly out (left), I am pre loading my hips. I step out on the toes, and plant my foot as I shift weight and torque hips The technique is almost a wave of motion starting with foot and traveling through hips to upper body/technique.
Seriously: it does not matter.
What does matter is the intent of the strike and the capability of the student. And as to the intent of the strike, remember, not all "striking techniques" are meant for striking - like the spearhand, for example. While I do break with the spearhand, that is solely for show and points, not for actual usage in combat.
While hip rotation is important in some breaks, it is less important in others (eg, some downward or low strikes).
Nevertheless, we need to remember what it is that we are breaking for. If for power or speed demonstrations, then by all means, choose the method that will allow to build up potential energy in the torque of the hip twist (so, usually same-side foot forward). Just remember that the follow through won't have that opposite side foot forward - that can throw you off balance.
But if the purpose of the break is to demonstrate balance, technique, and accuracy, then think about it: the technique you're using for the break applies to the technique you'll use in self-defense. And in self-defense, do you think you will adjust your stance to the technique you're going to use? No, you're going to use the technique in any stance you happen to be in - that is the ultimate goal, isn't it?
In that case, it does not matter the stance. Show you can issue the technique in any stance - even a crane stance if you want.
Depending on the break, for example when I use a downward inside knifehand (ridgehand), I might use same side foot forward - not because I need the twist of the hip, but because the other way I can't easily follow through.
If instead, I were throwing a horizontal ridgehand (so, out-to-in x-axis technique) then I'd want to stand reverse foot forward. Yes, I lose out on the twist, but I don't need that much twist to go through 2 or 3 boards - and any more boards is dangerous (and useless) for a ridgehand. So, with my opposite foot forward, that helps me keep my balance through the break. With same side foot forward, and if I have a clean break through, there could be a tendency to fall forward and not have that foot forward to break that fall. I might even opt for a horse stance, balancing the need for torque vs maintaining balance. Which I choose might depend on my mood. For a beginner, I'd probably have them use a horse stance, since they probably won't be piling up the boards.
With punching - a Z-axis technique - the reverse punch allows the rear foot to absorb the equal-and-opposite force the board imparts on the fist, and with the foot firmly planted on the floor, the result is a much better break for the breaker. I would recommend this method for the competing power breaker, or for the beginner who is nervous about breaking on his or her first rank test. The forward punch is, to me, much less stable.
Again, it depends on the student. I don't have students doing power breaks as a means for demonstrating the balance, speed, accuracy, etc part of their techniques, because it detracts from what the techniques can be used for. For junior students, I don't encourage power breaking. For advanced students, I let them do what they want.
Rotating/thrusting the back hip forwards is - when done properly - a stronger movement than rotating the front hip inwards. This video of Kagawa-Sensei explaining how to get proper hip rotation shows how the full strength of the back leg's quadriceps and glutes directly thrusting against the ground produces a strong, snappy forwards motion of the back hip: that explosive power can be used in a reverse elbow or reverse inward knife hand.
If you use the front hand, your rotation is powered by a much weaker hamstring contraction in the front leg, and can be further restricted by the friction between the front foot and floor, which when dragging the front hip inward for an obverse strike is more prone to slip than the reverse technique where the back leg thrusts at a greater angle down against the floor.
The LiveStrong website explains:
The normal hamstring to quadriceps strength ratio is between 50 and 80 percent, with 100 percent being equal strength, according to research.
(Of course the amount of weight the muscles can lift isn't directly correlated to the amount they can contribute to an explosive strike, but the above fits in with the quads being much stronger generally.)
OP's commented on John's answer: "...[I] notice a slowing down of the rotation by the time I get much past 45 degrees past the board." Once the board is broken, it doesn't matter if you slow down. Each technique has a depth range at which it's focused and powerful; you want to hit the target while inside that range. The upset punch (fist pointing upwards and travelling horizontally into the opponent's mid section) is a great example of a technique with very limited range: if you're over-extended - too far from the target - say your elbow is already passing your hip at the time you make contact - your technique will be ineffectual. You need to hit with it cleanly inside the range and then should be recovering balance and position for further defence or attack, not following through for the sake of it.