Process oil, basil, mint, garlic, and pepper in a blender until smooth, about 15 seconds. Pour mixture over Halloumi in a medium dish; stir to coat. Cover and chill 1 hour.

Preheat grill to high (450°F to 500°F); or heat a grill pan over high, and lightly grease. Add tomatoes to cheese mixture; toss to coat. Thread tomatoes, zucchini, squash, and cheese onto 8 (8-inch) skewers; sprinkle with salt. Place kebabs on lightly greased grill grate or in grill pan. Grill, uncovered, until cheese is lightly charred and vegetables are tender, about 1 1⁄2 minutes per side. Garnish with basil and mint leaves.