I have several Hampton Bay ceiling fans whose remote controls have a REV (reverse) button. One of the fans however the REV function does not work. The remote is sending the reverse signal - I hear a clicking noise in the motor and the fan comes to a stop, but instead of reversing, it simply starts back up in the same direction (with another clicking noise).

I live in a hot environment and I only need the fan to go in one direction - blowing down.
QUESTION: If I switch the wires going to the motor (Live and Neutral), might that cause the fan to spin in the desired (opposite) direction?

Reversing hot and neutral will not reverse Fan direction. There is a pair or wires that can be switched, due to the numerous colors used and lack of manufacturers drawings, it is difficult.
If the colors were the same on this and another fan, unplugging fan from receiver(on your good reversible fan. Then several tests of the switching(forward and reverse), hot, neutral, and continuity will allow you to see what wires to reverse. Don't know if replacement receiver is available. Ceilingfan may know.
Good Luck.

There is a trick posted somewhere on these forums on how to reverse the fan. It involves stopping the fan, spinning the blades by hand, and starting it again.

Reversing the wires won't work.

To get it to operate fully you will need to replace the receiver.