Would a vertical stabilizer that extends both above and below the centerline be benefit or hindrance?
As long as the vertical tail surface sits above the centerline (more precisely the longitudinal axis of inertia - for now we can assume both fall together), any side force will also create a rolling moment.  This is undesirable because now a yaw command will not only create the intended yawing moment, but also a rolling moment.

If you want to yaw in order to turn, the vertical tail above the centerline will even roll the aircraft in the wrong direction, so you need more coordinated aileron command than with a symmetrical vertical tail.

So the short answer to your question is: Terry is right: Extending the vertical tail below is clearly a benefit.

This can be seen by the widespread use of ventral fins in combat aircraft. Adding area below the centerline helps to reduce the roll contribution of the vertical tail in a sideslip, which improves handling characteristics (more directional stability AND lower yaw-induced roll). See the picture below for an example where the effects of additional vertical tail area are compared to the effects of a ventral fin of the same area.



At supersonic speeds, the low aspect ratio ventral fin has a nearly constant contribution to directional stability, whereas the wing-like vertical tail loses effectivity in proportion to the Prandtl-Glauert factor $\frac{1}{\sqrt{Ma^2-1}}$. The small ventral fin of the F-104 raised directional stability by 30% at Mach 2.



At high angles of attack, the vertical tail is in the wake of the fuselage which reduces local dynamic pressure and, consequently, effectivity. The ventral fin then is in ideal flow conditions, so it can help to stabilize the aircraft at high angle of attack, just when the fuselage contribution to instability is largest.