Example 7 As shown in the figure, in the right triangle $\triangle ABC$, it is known that $BC=a$. If the line segment $PQ$ of length $2a$ has point $A$ as its midpoint, for what value of the angle between $\overrightarrow{P Q}$ and $\overrightarrow{B C}$ is the value of $\overrightarrow{B P} \cdot \overrightarrow{C Q}$ maximized? Also, find this maximum value.