Place eggs in medium-size saucepan. Cover with water by 1 inch. Bring just to boiling. Remove from heat. Let stand 15 minutes.
Meanwhile, combine pickle juice, lemon juice, regular and reduced-fat mayonnaise, celery seed, red pepper, salt and black pepper in medium-size bowl. Add carrot, radish, pickles and tuna; stir to break up tuna.
Pour off hot water from eggs; run under cold water until cool to touch. Peel eggs; coarsely chop. Add to tuna mixture. Serve with shredded lettuce and pitas or English muffins.