If you type into the browser address bar the absolute URL to one of the image files used by the Add/Edit Appointment form dialog, such as
http://yoursite.com/ig_common/20083CLR20/Forms/WebSchedule/Images/attach.gif (the paper clip button image)
are you able to access it successfully without the Windows "Connect to" dialog box? This HTTP request doesn't involve the database server at all, it will just check if the permissions/config of the WebSchedule Forms Web Application are causing the prompt for authentication to appear.
If you're asked to authenticate to access this image file, then you should check that the WebSchedule Forms folder in IIS has been configured as an Application and that users who can access your ASP.NET application that is using the WebSchedule are also authorized to access the WebSchedule Forms, that WebSchedule uses for its dialogs, etc.
You can also try testing it the other way, disconnect the WebScheduleInfo from the WebSchedule data provider (temporarily of course), and see if you can add an Appointment in the UI without a data connection. Of course, this Appointment won't persist anywhere, but if you are able to add the Appointment in the UI (the Add Appointment dialog appeared correctly without displaying a "Connect to" dialog box for authentication) then your WebSchedule Forms are set-up properly ... and it more than likely is something related to how your database connection string is handling its authentication.
The variety of security/configurations possible make it hard to identify what is causing the prompt for authentication. But these two experiments should help you to isolate whether it is the setup of the WebSchedule Forms in IIS, or the database connection.
HTH