Prepare grill for cooking. If using a charcoal grill, open vents on bottom of grill.
Rinse squid under cold running water and pat dry with paper towels. If squid are large, halve tentacles lengthwise and cut longer ones, if attached, crosswise into 2-inch pieces. Pull off flaps from squid bodies, if attached, and reserve.
Cut each squid body lengthwise along "seam" and open flat. Score large squid, including flaps, lengthwise with a sharp knife, making cuts close together (about 1/8 inch apart like ridges on rigatoni) and being careful not to cut through squid, then halve squid bodies crosswise.
Squeeze enough juice from 2 lemons to measure 1/4 cup.
Cut remaining whole lemon into wedges.
Whisk together lemon juice, oil, garlic, 1/2 teaspoon salt, and 1/8 teaspoon pepper until combined well and transfer 1/4 cup dressing to a large shallow bowl, reserving remainder.
Add squid to dressing in shallow bowl, tossing to coat, and marinate at room temperature, turning once, 10 minutes.
Thread 2 skewers through each squid body, near opposite sides of body (to keep squid flat during grilling). Thread all the flaps onto 1 or 2 skewers. Thread all tentacles crosswise (through thickest part) onto skewers, without crowding, letting tentacles dangle. Discard marinade.
When fire is hot (you can hold your hand 5 inches above rack for 1 to 2 seconds), grill squid on a lightly oiled grill rack, uncovered, turning once, until they just turn opaque, 1 to 1 1/2 minutes total.
Transfer to a cutting board and remove skewers.
Cut squid bodies lengthwise into 1 1/2-inch-thick strips. Halve strips crosswise if using larger squid.
Toss squid with 3 tablespoons reserved dressing and salt and pepper to taste in a clean bowl. Toss arugula in another bowl with just enough of remaining dressing to coat, then season with salt and pepper. Divide arugula among 4 plates and top with squid.
