Easy to fix.

Remove the  spare and you will see a box surrounding the shifter
linkage.  Reach in from the back with grease on your fingers and apply
it to the left side of the box and around the linkage.  The squeak is
usually from the linkage rubbing the side of the box, which provides the
restriction of side to side movement of the shift lever.

When this squeak starts usually your upper front control arm bushings
are squeaking too.  I have an easy fix for this too if anyone has this
problem.

Stuart

'84 Westy, 196k
'65 MGB
'73 MGB GT