Can botox to the 11s only cause eyebrow or eyelid droop?

More than likely your eyebrow droop was from the location of your botox and improper layer as well.  Medial brow ptosis or descent of the central brow can occur for several reasons:
Botox placed too low within the frontalis muscle along the forehead including central forehead.  The frontalis muscle elevates the forehead and by decreasing activity here the central brow can drop
Botox placed too superficially along the corrugator.  The botox will again impact the frontalis muscle and paradoxically lower the central brow.
Botox is best injected by an experienced injector to lessen wrinkles while improving facial appearance.

The easiest way to improve your results after treatment with Botox is to seek out a better injector.  Inform your surgeon about your past issues, and he/she should be able to tailor your treatment to achieve a much more attractive result.  

Thank you for the question. The eyebrows can fall if Botox in injected too low in the forehead. Inexperienced injectors frequently cause this problem. Injecting the area between the eyebrows (the 11's) is less likely to cause ptosis; however, occasional the Botox can migrate to the upper eyelid and cause ptosis. Fortunately, the drooping eyelid usually resolves within a month. Make sure you see an expert injector. Good luck.