Pamela from Woebot here! We plan to develop non-FB options in the future, and they could involve entirely client-side stored data with no internet relay. There's a mailing list here if you'd like to find out when those are offered:
https://www.woebot.io/#besides-facebook
On the other hand although the creators of the bot have good intentions I wonder about the fact that you are using the internet to relay personal medical information. Also, I don't know if your woebot information is stored anywhere so if Woebot gets hacked a bunch of PII relating any medical data you send. I wish that this was a standalone app that would prompt me to upload my data if and when I see fit.