I have a white fiberglass shower that has become very discolored after
using Arm & Hammer Clean Shower to prevent soap scum and mildew
stains.  I am trying to find a cleaning solution that will work to
remove the stains so that I do not have to replace or paint the
shower.

The shower has a brownish, almost rust color in it now from using the
Arm and Hammer cleaning solution.

Maybe this can help:
http://www.stretcher.com/stories/05/05jan31a.cfm
http://www.diynetwork.com/diy/lv_household_tips/article/0,2041,DIY_14119_3001762,00.html

I was also thinking of vinegar and baking soda, having read that they
will remove the yellow tinge from enamel sinks.

But you seem to have a more serious problem.  Is there a lot of iron
in your water?  If you do, I don't know if it will get past the
surface, causing a permanent stain.  I hope not.