Yes bleach will usually unclog a sewer or shower drain depending on what's clogging the shower drain.
Some more severe clogs may require snaking the shower drain or taking the drain apart to get at the clog.
But you can pour undiluted bleach down the drain and let it sit for 10 to 20 minutes to let the bleach work and then run some hot water down the drain if it clears.
You can also use baking soda and Vinegar as a natural chemical free way of clearing clogs in the drain.
Pour some baking soda down the clogged rain and then pour some vinegar into the drain and cover the drain with a drain plug or put something over the drain.
Let the baking soda and vinegar sit in the drain for 15 to 20 minutes and then remove the cover or drain plug and pour some boiling hot water down the drain.
If you have hair clogging the drain the baking soda and vinegar will break down the hair that is clogging the drain and will also clear some other clogs.