The brake wiring and the rear turnsignal wiring both do go through the turnsignal switch, so it is a possibility.
If you want to check the switch, the best thing to do is go down to the plug at the bottom of the steering column, and poke the wires with a meter.
Make sure the turnsignal switch is in the center position.
Put one of your testlight or meter leads on ground, and then probe the lightgreen wire. While you have the lightgreen wire on the meter, push the brake pedal. You should get 12v, this is the brake signal wire going into the turnsignal switch.
If that's good, then probe the orange/lightblue wire. When you push on the brake, you should get 12v, this is the right rear brake turn wire.
If that's good, then probe the lightgreen/orange wire. When you push on the brake, you should get 12v, this is the left rear brake turn wire.
If all these tests pass, and your brake lights still don't work, your problem is going to be in the wiring to the rear or some other problem at the rear.