As a follow up to my problems with the squealing speedometer, Mike was right all along. If you take the time to take the dash apart, it would be better to send the instrument out the first time for repair or overhaul.
After I got the speedo out, I attempted to lube the cable and also lube the inner shaft at the speedo housing. I tried to check the speedo with a variable speed drill (in reverse), and found that the speedo now was erratic and caused the needle to jump. Being an optimist, I reinstalled the instruments in the dash and took a test drive. The speedo was very erratic for the first mile or so. After a couple of miles it began to settle down to a steady read out. I thought that I had fixed it up until about mile seven when it started up squealing once again. I opened up the hood and moved the speedo cable a couple of times and the squealing stopped. However, each time I did this the noise would reappear in short order.
My conclusion is that either the speedo cable at the housing is binding, or the internal mechanism of the instrument needs lubrication or has worn parts.
I will be taking the dash apart once again and this time sending the speedo out for repair.