Heat the oven to 375ºF. Cut each squash in half through its equator and scoop out the seeds. Cut the halves into ¼-inch-thick slices and place them in a single layer on a rimmed baking sheet. Sprinkle the oil, thyme, salt, and pepper over the squash and toss to coat it with the seasonings. Roast until tender, 10 to 12 minutes.
Heat a grill, grill pan, or panini press to medium. Spread the mayonnaise and mustard over four slices of the bread. Arrange the cheese on the other four slices and then layer on the squash, pepper pieces and arugula. Top with the spread bread and coat the sandwiches all over with cooking spray.
Place them on the grill or grill pan, compressing with a heavy weight, such as a cast-iron pan or foil-wrapped brick. If using a panini press, close the lid. Cook the sandwiches until nicely marked by the grill and the cheese melts, 3 to 5 minutes per side, turning 90 degrees halfway through cooking on each side to make crosshatched marks. Cut each panini in half before serving.
Tip: If you can’t find delicata or sweet dumpling squash, use half of a medium butternut squash instead. To soften the skin, prick it all over with a knife and microwave the squash on high for 2 minutes. Then peel it, seed it, and slice it into ¼-inch-thick rounds. It may require 3 to 5 minutes more roasting time than the delicata needs to get tender.