If $a > 0$, the distance from the vertical line with equation $x = a$ to the $y$-axis is $a$. If $a < 0$, the distance from the vertical line with equation $x = a$ to the $y$-axis is $-a$. In each case, there are exactly two points on the vertical line with equation $x = a$ that are also a distance of $a$ or $-a$ (as appropriate) from the $x$-axis: $(a, a)$ and $(a, -a)$. These points lie on the horizontal lines with equations $y = a$ and $y = -a$, respectively. If the point $(a, a)$ lies on the line $3x + 8y = 24$, then $3a + 8a = 24$ or $a = \frac{24}{11}$. If the point $(a, -a)$ lies on the line $3x + 8y = 24$, then $3a - 8a = 24$ or $a = -\frac{24}{5}$. The sum of these values of $a$ is $\frac{24}{11} + \left(-\frac{24}{5}\right) = \frac{120 - 264}{55} = -\frac{144}{55}$.