Combine vinegar, water, butter, Worcestershire sauce, garlic salt, pepper, sugar, and minced garlic in a pot. Bring to a boil. Remove from heat and let marinade cool to room temperature, at least 30 minutes.
Place chicken in a resealable zip-top bag. Pour marinade over chicken, seal, and marinate in the refrigerator, 8 hours to overnight.
Preheat an outdoor grill for medium-high heat and lightly oil the grate. Transfer chicken pieces to the grill, shaking off excess marinade; discard remaining marinade.
Grill chicken until no longer pink in the center, 30 to 45 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).