You probably need to clean the splines. Remove the rear tires, and pop loose the nut at the hub, and remove it, and clean out the rust best you can and grease it up really well. Should solve your problem.
the most probable cause of clicking on rear brakes are weak return springs or a broken one.
Check ABS system.
heat/ac actuator
If the clicking noise you are hearing from the rear panel in your Mountaineer is coming from the passengers side, you could be hearing the relay contact opening or closing for the automatic headlights (if equipped), rear window defrost, rear wiper, or a host of other features. Behind the trim panel in the rear passengers side are a bunch of the control modules, including Security and Body Control.
Probably a damaged gear or bearing.
Chances are you have a rock(S) stuck in your rear tire tread. pry it out.
Apparently it is the rear air "blender" ( whatever that is(?)) My 2003 Explorer was making the loud clicking sound from the rear driver side above the rear left wheel. I lived with it for a month before I just found the answer on the net. I turned off the rear fan control (from the front controls) and the clicking has stopped. I guess there is no more air from the rear now but the clicking has stopped- Car is long been paid off and I'm happy again.
If it is in the rear end you need to change the third member oil.
I have same issue except I am on cold and get clicking when attempting to go to heat. also rear areas have heating or cooling
rear air unit has a air blending door solenoid, and is probably out. or broken making a clicking noise
Have you checked your Exhaust Muffler and pipes ? Probably the fuel pump
Bad strut or strut mount? Check for loose undercarriage components