Sidewalk and blackboard chalk does have wax and other additives in it that makes the chalk sticks glide more easily over the surface and to keep them from breaking. Carpenter's chalk has additives in it to make it more visible and to stick better. Over a period of time, these additives may create a shiny spot on the mating surfaces resulting in squeaks and slippage....the reason we chalk our box calls in the first place. While they do not work as well as chalk meant for use on box calls, they do work if one cannot find anything else. For box calls the driest and cleanest chalk work the best.
Some of the newer box calls work without having to be chalked and are better in wet weather than the old style. I still prefer the tone and sound quality of the old calls tho and keeping them dry, chalked and sanded correctly, are just part of the deal.
On sale as of this writing for $2.39. I bought one 20 years ago and there's still most of it left. One box will last a lifetime. Why bother with something that might work?
While they are all called "chalk", not all "chalks" are made from the same type of calcium deposits. Rock climbing and gymnast chalk are made to be more water(sweat) resistant than other types of chalk. Chalkboard and sidewalk chalks are made from softer materials. Both may or may not contain other additives that may deter the chalk from doing it's job on wood calls. I know of some turkey hunters that swear by pool chalk.
Only way to find out for yourself is to try it yourself. If it doesn't work, lightly sand it off and try something else. Like natman, I bought a stick of call chalk some 20 years ago and still have most of it left.