Hmm. In every other case in which it matters, double weapons are treated like two separate weapons - they must have the masterwork fee paid for each head, they must be enchanted separately, and they are treated as two weapons when they are fought with.
While your statement above is technically accurate with regards to the wording of the spell, it is not consistent with the treatment of double weapons in other cases. To be consistent, each head of the quarterstaff is treated as a separate weapon - so the shillelagh spell (and other boosting spells like magic weapon) would affect only one end of the staff. The other end is unaltered, and can still be used to attack for 1d6 damage.
This solution has the additional benefit of not incurring disbelief rolls from your players: "Gee, I just cast this spell, and now I suddenly can't attack with the other end of my quarterstaff..."
J