In he extreme cold the Brake light switch will turn on as if you're pressing the brake pedal. You will have to adjust the switch to back it off a little bit. Hope this helps.

Water in your brake fluid expands when it freezes increasing the pressure on you brake fluid causing your brake lights to come on. Drain all of the brake fluid and replace it. This is not a job for a shade tree.

Water gets into your brake fluid because the fluid gets warm when the brakes are used, then cools off. It is called condensation.