The so-called "military box pleat" is simply a box pleated kilt made from a nominal 8 yards of cloth (though in reality it was not uncommon to see these kilts made with much less cloth). In a four yard box pleated kilt, there is generally a good balance in the pleats. Assuming an average size sett repeat for the tartan, and an average sized gentleman, the two sides of each pleat will just meet in the center of the pleat, meaning the pleats running left and right will both be of equal depth.

Now, when you increase the amount of yardage used in a box pleated kilt, the pleats get smaller, and typically what this means is that the pleat depth on one side gets more narrow, while the depth increases in the other direction. So if you are making a kilt from 7 or 8 yards, the pleats end up looking very much as you have described -- "like knife pleats with the ends turned back."

This style of pleating was indeed worn by a number of military regiments, notably the Seaforth Highlanders and the Queens Own Cameron Highlanders. It was a hold over from the days when all military kilts were box pleated (pre-1854). After the mid-nineteenth century many regiments switched to the newer box pleats. You also saw a general increase in the amount of material used in making the kilts. So those regiments who retained the traditional box pleat kept the original pleating style, but increased the yardage following general fashion.

The box pleated kilt also fell out of general fashion by the end of the nineteenth century (though you will still find it mentioned in some turn of the century accounts), and so people began associating the style strictly with those regiments who retained the practice. Thus box pleats came to be thought of as a "military thing" although there really is no more reason to associate box pleats with the military than knife pleats.