Can I use brake fluid or steering fluid to stop a fan from squeaking?
Brake Fluid is not a lubricant, but a glycol based hydraulic fluid designed to have a high boiling point and to absorb water to prevent corrosion (why the brake system should be completely bled out every so often).

Power Steering fluid and Transmission fluid are petroleum based hydraulic fluids, more useful for their ability to transmit pressure, resist heat and work with fluid filled clutches (friction systems) and are not general purpose lubricants, except in the special case of the systems for which they are designed.

You'd almost be better off using engine oil, but its detergent properties are not useful for this type of operation.

Any box store will have 3in1 or turbine oil which are more appropriate for lubricating the shafts on an electric motor.