Thanks for the suggestions.
I don't think there's much glue between the blocks to scrape out - the box
had been stored I think for years in a hot closet and when I was given it
there were lines of either glue or varnish on the surface along the seams of
the parquet, as though the glue had melted and run out there.
The squares had also probably shrunk as they dried out - there was no
varnish on the underside of the lid, just bare wood. Just having it in a
more humid environment (I live on the coast and it's been raining, humidity
way up) has made the blocks fit much more tightly. I'm considering either
just seeing if tapping the pieces into place and oiling it, and hoping the
oil will keep the wood tight enough, or if a cyanoacrilate glue is thin
enough to run into the seams and will hold on the oily teak, using that and
applying the oil.
Another question: it looks like the box was originally built without any
feet - just a flat bottom - but someone added what look to be four
thumbtacks to raise it, and one of those is now missing. I'm thinking of
replacing the tacks with either some little brass feet, if I can find some
suitable, or more likely, to apply some adhesive felt-pad feet. I am
wondering how well they will stick to the oily teak bottom, and if I apply
something like the Watco teak oil finish, is it going be to even less
satisfactory attaching self-stick pads like that?
The bad news is, the box appears to be just the right size for the cigars I
infrequently smoke. I don't smoke cigars enough to warrant owning a
humidor, but I'm afraid if I have a box so suitable for use as a humidor, I
might smoke them more often.