In a medium bowl, mix all the ingredients for the spice rub together until well blended; set aside.
Cut the zucchini and squash in half lengthwise and gently scrape out the seeds with a spoon. Place the vegetables skin side down on a sheet pan. Sprinkle them evenly with salt and let them sit for 5 minutes to sweat. Rinse them in cold water to remove the excess salt and pat them dry with paper towels. Slice them widthwise into 1-inch pieces. Using the sharper end of a chopstick, gently screw a hole through the center of each piece. Put them into the spice rub and let them marinate while you prep the carrots.
Using a peeler, peel down the baby carrots to the width of a chopstick, leaving the top inch of the carrot to use as a handle. Gently push a piece of zucchini onto the carrot with the skin side facing the "handle" of the carrot. Next skewer a piece of squash at a 90 degree angle to the zucchini. Repeat 2 more times so that there are 2 pieces each of zucchini and squash on the carrot "skewer." Repeat until all the carrots have been used.
Preheat the grill to medium-high heat. Place the skewers on the grill and cook for 3 minutes on each side or just until the vegetables are tender. Remove them to a platter and serve.