You may have a faulty brake light switch. Look up under your dash at the shaft attached to your brake pedal. Somewhere up the shaft should be a flat surface attached to the shaft, and a corresponding plunger-like switch. Fiddle with the plunger (which must slide out to light the lights, and in to shut them off) to see if you can change things. If the plunger seems stuck, that may be your problem. They run about $30 or so at the parts counter at the dealership, and are usually only held in by one bolt, and a single plastic connector. Easy to DIY.
New brake switch did not fix the brake lights out...there is a definite clicking sound when the brakes are applied though. This clicking sounds like it is coming high up on the steering column, almost like under the steering wheel.