Let $ w = \frac{z_2}{z_1} $. Then $ w $ is pure imaginary, so $ \overline{w} = -w $. We can write \[\left| \frac{2z_1 + 7z_2}{2z_1 - 7z_2} \right| = \left| \frac{2 + 7 \cdot \frac{z_2}{z_1}}{2 - 7 \cdot \frac{z_2}{z_1}} \right| = \left| \frac{2 + 7w}{2 - 7w} \right|.\]The conjugate of $ 2 + 7w $ is $ \overline{2 + 7w} = 2 + 7 \overline{w} = 2 - 7w $. Thus, the denominator $ 2 - 7w $ is the conjugate of the numerator $ 2 + 7w, $ which means they have the same absolute value. Therefore, \[\left| \frac{2 + 7w}{2 - 7w} \right| = \boxed{1}.\]