Lightly coat an unheated panini griddle, covered indoor electric grill, or large nonstick skillet with nonstick cooking spray. Preheat over medium heat or heat according to manufacturer's directions. Add chicken. If using griddle or grill, close lid and grill for 6 to 7 minutes or until chicken is no longer pink. (If using skillet, cook chicken for 10 to 12 minutes or until chicken is no longer pink, turning once.) Cool chicken slightly; split each chicken piece in half horizontally and cut crosswise into 2-inch-wide slices.
In a small bowl, combine dried tomatoes and boiling water. Cover and let stand for 5 minutes. Transfer undrained tomato mixture to a small food processor (if you have a larger food processor, you will need to stop and scrape down sides occasionally). Add roasted red sweet peppers, balsamic vinegar, oregano, large clove garlic, and ground black pepper. Cover and process until smooth.
Spread the Dried Tomato-Pepper Spread on cut sides of bagel bread squares. Place chicken on bottoms of the bread squares. Using a vegetable peeler, cut very thin lengthwise strips from the zucchini. Place zucchini strips on top of the chicken. Place bagel square tops on top of the zucchini, spread sides down. Press down lightly. Lightly coat the top and bottom of each sandwich with nonstick cooking spray.
Place sandwiches on griddle, grill, or skillet, adding in batches if necessary. If using griddle or grill, close lid and grill for 2 to 3 minutes or until bread is toasted. (If using skillet, place a heavy saucepan or skillet on top of sandwiches. Cook for 1 to 2 minutes or until bottoms are toasted. Carefully remove saucepan or top skillet; it may be hot. Turn sandwiches; top again with the saucepan or skillet. Cook for 1 to 2 minutes more or until bread is toasted.)