In a bowl, using a wooden spoon, crush the spearmint with the sugar and lime quarters until the sugar starts to dissolve and tastes like mint. Add 1 cup of the club soda and stir until the sugar dissolves completely. Strain the mixture through a coarse sieve into a pitcher. Stir in the lime juice and rum. Fill 8 tall glasses with ice and add a mint sprig to each, lightly crushing it with your hands. Pour the mixture over the ice, top each glass with club soda and serve.