Place wood chips in a bowl and set aside.
Pop tabs on beer cans and make 2 additional holes in each top using a church-key opener.
Pour half the beer from each can over the wood chips. Add additional beer or water to cover chips completely.
Soak wood chips for 1 hour, then drain.
Set up grill for indirect grilling and preheat to 350°F.
Sprinkle 1 teaspoon barbecue rub in neck cavity and 2 teaspoons in main cavity of each chicken.
Add 1 tablespoon barbecue rub to each half-full beer can (foaming is normal).
Season outside of each chicken with 2 tablespoons barbecue rub.
Stand beer cans on work surface. Hold each chicken upright and lower over can so it goes into main cavity.
Pull chicken legs forward to form a tripod base. Chicken should sit upright and stable.
Carefully transfer chickens to grill center over drip pan, away from direct heat.
Add wood chips: if using charcoal, toss half on each coal mound; if using gas, place in smoker box.
Grill chickens for approximately 1½ hours, maintaining 350°F temperature.
Cook until internal temperature reaches 165°F in thickest part of thigh.
Carefully transfer birds to platter in upright position.
Lift birds off cans to carve and discard cans.