Will broken toilet wax ring cause upstairs bathroom waste line to overflow when shower is running?

The purpose of the wax ring is to kept the toilet from leaking from underneath toilet on floor when toilet is flushed.For a waste line to overflow you have a blockage somewhere.The wax ring could be leaking due to the blockage.

Pffmugs is correct. First wax rings do not break, they can fail but its soft and mis-shapened. You can a clog downstream of the toilet and the shower. If nothing else backs up the clog is not far downstream. Where exactly do you see the backup, in the shower drain?

The original poster has never returned even after 11 months. We never pinned dow the real issue and I doubt the poster will see your link. Wax rings are cheap, easy to find and not difficult to install but they do not cause backup. Thanks for the link

Could this be one of those "unanswered" old posts?

No Tom asked and answered twice the same day. The old post being dragged back to life are unanswered.