My 2006 chevy HHR Lt's break and signal lights wont work, brand new BCM didn't fix the problem, also no sound when turn signals are on. Checked all fuses

 

Why a new BCM ? My guess would be fuse, relay, wiring, turn signal switch or bad ground.
If you don't have working turn signals , then there is not going to be any "sound". FYI. That "sound" normally comes from the left front speaker. Of course if you have a bad speaker, then there will be no turn signal sound either. Or, if the stereo has been replaced and the installation didn't include the chime adapter, the again, no turn signal sound.

 

And if the new BCM was not programmed, good luck.
Is it ONLY the lights in the rear? Does that include the high mounted brake light?
What about the marker lights?
Any recent work before the BCM?
Did you check the bolts that hold the under hood fuse box together, they need to be really tight. Fuses and relays tend to work loose, check them all.