Possible that Juvederm under the eyes can cause prolonged and excessive watering/tearing afterwards?

It is unlikely that your Juvederm treatment is causing your prolonged eye watering. If it persists, I recommend following up with your Doctor. If necessary, Juvederm can be reversed with hyaluronidase. It may take a series of treatments. See an expert. Best, Dr. Emer

There is a small possibility you could be having an allergic reaction to a Hyaluronic Acid product.  Although rare, patients may sometimes react to the product or rather one of the side chains which hold the  product together.  I would suggest dissolving the product.

Thank you for sharing your question and I am sorry to hear of your excessive eye watering/tearing.  Though Juvederm can cause prolonged swelling in the lower eye region I do not know of it causing difficulties with tear production, especially at a week after treatment.  If the watering continues you can return to your injector and have the Juvederm dissolved to see if this improves your symptoms.  Best wishes.