According to Gauss's Law, the total flux through a closed surface enclosing a charge $q$ is $\phi = q/\epsilon_0$. A charge placed at the corner of a cube is shared equally by 8 identical cubes arranged around that corner (4 in the lower layer and 4 in the upper layer). Therefore, the portion of the charge effectively enclosed within one specific cube is $q_{enclosed} = q/8$. Consequently, the electric flux through this cube is $\phi = \frac{q_{enclosed}}{\epsilon_0} = \frac{q}{8\epsilon_0}$.