I tried this on Mathematica 8.0.4 on a Mac and it shows the same problem (missing vertical lines at x={60,80} also.
What version and OS did you have your problem and successes on?
You can try giving explicit Exclusions:
Plot[g[x], {x, 0, 10}, Exclusions -> {x == 6, x == 8}, ExclusionsStyle -> Red]
On my system (Mma version 10) I get the vertical lines also with this:
Plot[g[x], {x, 0, 10}, ExclusionsStyle -> Red]
Just add the option to the plot: Exclusions -> None
g[x_] := \[Piecewise]{{20, x < 6}, {-60, 6 < x < 8}, {0, True}} Plot[g[x], {x, 0, 10}, Exclusions -> None]
