Pour 1/4 to 1/2 inch of sugar onto a small, shallow plate. Run one of the lime quarters around the rim of each cocktail glass, then dip the glasses into the sugar to rim; set aside.
Squeeze all of the lime quarters into a sturdy glass pitcher. Toss the juiced limes into the pitcher along with the mint, strawberries, and 1 cup of sugar. Crush the fruits together with a muddler to release the juices from the strawberries and the oil from the mint leaves. Stir in the rum and club soda until the sugar has dissolved. Pour into the sugared glasses over ice cubes to serve.