Wiring help for a Switch with LED Guide Light
Yes, the white wire is required for the light built into the switch to work.  It does not affect the switching of the bathroom light.  You will need to create a pigtail (a short piece [6-8"] of wire connected to the two in the box) to join to your switch.

In you picture, the red wire going to the switch is the load, and the black wire is the hot.