What's a neck square?

A cravat or ascot? No.

Not even one of those neckerchiefs that Prince Charles wears skiing. 

A Kenneth More thing; like pipe smoking and being a bluff, clubbable cove who always plays a straight bat.

I've never seen much practical difference between and ascot/day cravat and a silk scarf folded and worn like one. At least one style writer expressed the opinion that the scarf was much classier than the ascot and compared the latter to a pre-tied bowtie. I find the two hard to differentiate, at least in photos.

I don't like the looks of neckerchiefs worn with a square knot in the front unless they are being worn in this manner to protect the back of your neck from the sun. With the knot in the back, cowboy style they are okay, actually not that dissimilar to the ascot. Kind of a curious dichomotomy. The cowboy's bandanna is rugged, manly and outdoorsy, while the ascot is perceived by many as effete, pretentious and faux-patrician. According to eminent old-time gunwriter Jack O'Connor, quite a few of the Arizona cowpunchers of his youth favored silken neck scarves.