Bring first 7 ingredients to a boil in a saucepan over medium heat, stirring often. Reduce heat, and simmer 5 minutes. Spread 1/2 cup tomato sauce mixture in bottom of a lightly greased 13- x 9-inch baking dish.
Melt butter in a large skillet over medium-high heat; add chopped onion and green bell pepper, and sauté until tender. Remove mixture from heat.
Stir in chicken, Ranch dressing, and pepper. Spoon about 1/4 cup mixture just below center of each egg roll wrapper. Fold in left and right sides to partially enclose filling; fold up bottom edge, and roll up. Place, seam side down, in baking dish.
Pour remaining tomato sauce mixture over rolls.
Bake at 350° for 25 minutes. Sprinkle with cheese. Bake 5 more minutes or until cheese is melted.