My Snodgrass box rides in a soft camo pouch, and my new Billy Bush rides in a custom leather holster that also loops onto my belt.
Sometimes I will drop my long box into one of the big bellow pockets of the vest, here's how to keep it quiet:
Know those head lamps / flash lights with the stretchy fabric elastic straps that you wear on your head? I bought a really cheap one and cut the elastic strap off.
I then cut a about 3-4" long, and wrap it around the paddle at the center where it touches the sides of the box. I then put a couple drops of super glue on the strap after pulling it just tight enough to keep it snug on the paddle. Then I take a wide rubber band and double wrap that around the paddle and box handle.
When ready to call, pull the rubber band off and slide the elastic strap toward the screw end and call. That is far enough up toward the screw it never interferes with the paddle. When you are ready to move, slide it back down and put the rubber band back on. Never makes a squeak.
After awhile the elastic band picks up the chalk and becomes coated, which helps keep the box paddle chalked and ready.