Originally, square braid candle wicks were used to deal with the high viscosity of beeswax candles but they can also be used with paraffin wax and soywax to create large pillar candles, container candles and "novelties" candles. Their section is not actually square, rather round, but their thickness makes them able to stand up straight in a candle's melt pool. Square braid wicks are made of 100% natural cotton fibers and receive a special treatment to make them work better with highly viscous waxes.
They're thicker than their flat braid counterpart and can be used in pillar candles with a diameter of 125mm or more.
The classification system used by Candlewic ranges from 6/0 to 1/0, where 6/0 is the thinnest wick and 1/0 the thickest, then continues from #1 to #10, where #10 is the thickest (pretty much a piece of rope, used with outdoor terracotta citronella candles).
The difference between the two numbering schemes is that the wicks referenced #1 to #10 are braided much more loosely, which makes them thicker without making them heavier. Also note that in the 6/0 to 1/0 series, the biggest number refers to the thinnest wick whereas in the #1 to #10 series, the biggest number refers to the tickest wick.
Like the flat braid, the square braid curls slightly as it burns, allowing for a better disposal of the burnt part of the wick and reduced carbon buildup (mushrooming).
It is possible that your wick supplier doesn't use the number of ply to identify flat braid wicks but rather indicates the size and type of candles they are most appropriate for.