Trim fat from meat. Sprinkle both sides of meat with Italian seasoning, salt, and black pepper. If necessary, cut meat to fit into a 3 1/2- or 4-quart crockery cooker. Place mushrooms, onion, and garlic in cooker. Add meat. Pour undrained tomatoes and vinegar over all.
Cover; cook on low-heat setting for 7 to 9 hours or on high-heat setting for 3 1/2 to 4 1/2 hours. If using low-heat setting, turn to high-heat setting. Add squash or zucchini and sweet peppers to cooker. Cover; cook on high-heat setting for 30 minutes more.
Remove meat from cooker. Stir in drained artichoke hearts. Thinly slice meat across grain. Arrange meat on the focaccia. Using a slotted spoon, place vegetable mixture over meat. Drizzle with a little of the cooking liquid. Sprinkle with cheese. To serve, cut into wedges.