I did consider adding a clear all at some point, I guess it makes a lot of sense in a scenario like this.
I added a new parameter for delquote. You simply have to add &clear=1 at the end of the command.
I highly recommend setting it to Owner only, as anyone accidentally running that would delete all your entries.
It would look something like so:
$(urlfetch http://twitch.center/customapi/delquote?token=TOKEN&data=&clear=1)
Make sure to replace the token with the delete token, and again, set the access level properly.