Technically brake fluid can’t freeze solid like water due to the fact that it is an oil. But it can reach a point where it becomes so thick that it no longer becomes effective at transferring force from the master cylinder to the wheels. The fluid can “gel” or congeal. Base mineral oil has a “working point” of -22C or -30F. But, it will never turn into a solid.
Additives to the base oil can push the limit to -45F or lower. It the temperatures go lower chances are the pedal will be stiff, but not frozen. After a few stops, the heat of the brakes would warm the fluid. If you are driving in temperatures below -50F, you have a lot more problems than the brake fluid like engine oil and coolant.
Theoretically, the ABS pump on some vehicles could have problems if it activate and the brake fluid was thick enough. This could introduce air bubbles into the brake fluid. Also, since brake fluid is hydroscopic (absorbs water), the additional water can influence the viscosity at lower temperatures.
No, DOT 3,4 and 5.1 brake fluid will not freeze preventing you from stopping altogether. Chances are if you have experiences a hard brake pedal in the winter, it is condensation in the brake booster or vacuum supply hose that has frozen.