Process first 5 ingredients in a blender or food processor until smooth.
Place green onions in a shallow dish or heavy-duty zip-top plastic bag; pour lime juice mixture over green onions. Cover or seal, and chill 30 minutes, turning occasionally.
Remove green onions from marinade, reserving marinade; set aside.
Grill chicken, covered with grill lid, over medium-high (350° to 400°) 6 minutes on each side or until done, basting frequently with reserved marinade.
Grill green onions, covered with grill lid, over medium-high heat 3 to 5 minutes on each side or until browned. Serve with grilled chicken.