Could be the brake that is squeaking.  If so, polish the metal and
replace the brake felt or leather.  Be sure it is the take-up spool
squeaking; it could be a number of transmission friction points.

If it is the actual metal connection of the take-up spool then the
best lubricant (in my opinion) to use would be "Marvel Mystery Oil."
Remove the take-up clean off all the other stuff that has been applied
(use denatured alcohol) and apply "Marvel Mystery Oil."  If you have
ever used this on metal to metal friction applications you will never
use any other light weight machine oil.  You can get it at most
automotive supply stores.

Sam Harris
Greenville, North Carolina