EXPERT RESPONSE
If I understand correctly (and am making the correct assumptions), you have had someone develop a solution which renders your public folder calendars into an Outlook Web Access Calendar, and are exposing that to the Internet. In that case, it's the security settings on the IIS Server that hosts your OWA site, which controls challenge/response authentication. Working with the 'anonymous access' privileges should give you a way around users getting prompted.
That said, I would never recommend exposing Exchange data on the Internet without a firewall and appropriate security. It would be much simpler to figure out a way of having a developer write some code to provide a one-way export from the public folder calendar(s) to an IIS server in your DMZ, without actually exposing Exchange on the Internet, directly.
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Related information from SearchExchange.com:
Expert Advice: Password prompt when attempting to view a public folder
FAQ: Exchange public folders
Exchange Admin 101: Introducing public folders
Learning Guide: An administrator's guide to Exchange public folders
Reference Center: Exchange Public folder management tips and resources
Reference Center: OWA user authentication resources
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