You might try a 'light' spray of water on the serpentine belt, tensioner, and idler pulley while the engine is running and the noise is present.  If the sound of the squeal does not change, then the clutch / release bearing will become more suspect.

It is very difficult to diagnose a 'noise' remotely, and even more difficult to be accurate.

Unfortunately, servicing the clutch is one of the more labor intensive jobs on the MINI - you want to eliminate other possibilities as much as you can before you remove the transmission....

Ken

The car is at about 168,000 miles. I bought it used somewhere around 141,000. Since I've had it, I haven't had the clutch serviced, but it may have been done by the previous owner. They seemed to take decent care of it anyway.

As minimans mentioned, it 'may' be the clutch release bearing or the throwout bearing....

How many miles on the car and when was the last clutch service?

Ken

clutch release bearing.............

What about the idler pulley?

I own a 2003 Cooper S, and after a lot of nightmares I had to get the alternator replaced. About two weeks later a persistent squeal, similar to a belt squeal, started up and has not stopped. I took the belt off and went over each pulley but nothing seemed to be locked, catching, or even remotely inhibited. Recently, I've noticed that if I just rest my foot on the clutch, not pressing down with any force or engaging the clutch, but just resting it on there enough to push the pedal down half an inch, the squeal will get quieter or stop entirely. I've also noticed that if I sort of "jog" the clutch, the squeal may stop as well. 

I have no idea what could be causing it. I assumed clutch slip for awhile, but it squeals even when in neutral with the clutch pressed down all the way. Any thoughts or possible places to look?