Clutch Chirps Until the Pedal is Pushed in

Alright everybody, new to this forum and fairly new to the world of mustangs. Do not know car words in all entirely and just like simple work on my car. To start off, I have a 2000 v6 mustang that I bought a little over half a year ago. I started off not knowing how to drive a stick and learned fairly quickly with this little car. In about July, the car started chirping and was told it was the throw out bearing. So, like most people would do, I called up a mechanic and had them replace the clutch and throw out bearing. Literally the next day the chirping came back and so since I had a warranty on the replaced clutch, I took it back to the same mechanic. They double checked it and told me it was not the new throw out bearing nor was it the clutch. They told me it was a ball like item that engages the clutch, they did not know the official name of it but said it would be an item they had to special order from ford and it would cost a couple hundred to fix it? My main question here is, what would be the name of the item that is still causing that chirping and how much is that item.

Edit: Another question would be that, If I pulled down the Transmission myself to replace the Pivot Stud, should i replace the throw out bearing that came with the clutch kit the mechanic installed? If so, I've seen that the ford performance throw out bearing is great but is debated whether it would fit a v6 or not? Any advice on this would be great!!!

Okay, I guess another question would be, would these parts cause the car to chirp like it does?

When the mechanic that replaced my clutch went back in and checked for the chirping, he said they had to re-polish and shine a ball looking thing and that it was the cause of all the chatter, i am just confused on how the pivot stud, what I believe he is talking about, would cause that chirping?

I am 1000 percent positive that they changed out the old throwout bearing.