Should a exterior basement bathroom wall be insulated and have vapor barrier?(shower is on this wall)

A simple answer is yes. All exterior walls should be insulated.

Chuck

Agreed but the insulation is a bit tricky in your application. Normally the barrier goes directly under the drywall. In your outside wall the vapor barrier is on the basememt cement wall, then the insulations and finish surface. Do not use drywall or even greenboard. Cement board is good and use of a membrane like Shop Schluter Systems 3-ft 3-in x 16-ft -5-in Waterproofing Membrane at Lowes.com is a best practice

Thanks that's what I'll do.