You can tell in a few minutes whether this fix will be of any use. The offending part is protected by a large, squarish black rubber boot, and this has a slit especially for speedy regreasing with an aerosol can. With the engine off, stick the nozzle of a can of rubber safe grease into the slit, and give it a few squirts. If the clutch squeak vanishes for a few presses, then this is the fix for you.
I knew what I was looking for, thanks to forums in which people described greasing the ball joint. It's not clear exactly where to look at first glance, so I took a few photos for future reference. First a wide shot: look low down between the battery and exhaust headers. This boot is obscured by a thick hose, it's right next to the clutch slave cylinder.
Closer in: the boot is just to the left of the slave cylinder you can see circled in the photo below.
A view looking left from the slave cylinder - the rubber boot is circled in green, the greasing slit is highlighted in red.
Here's a view looking right from the battery side, showing more of the boot itself. Highlighted in green, and the greasing slit is highlighted in red again.
Squirt the aerosol spray in there , then give the clutch a go. If you're still hearing a squeak, then then this fix may not help. If you get a temporarily quiet clutch, great! A worn push rod is the proven culprit. In the next post, I have the part number to ask for/order over the net, and a comparison of old and new showing why greasing up the old rod isn't worth the hassle.