We have successfully migrated our organization from Exchange 2010 to 2013. The only remaining problem is a security dialog pops up when user attempts to connect to the public folder mailbox. If they type in their username (in domian\username or UPN format) and password, outlook will connect to the public folder database. It seems to be using basic authentication when it should be using NTLM. Autodiscover is working on everything but the public folders.

Before typing in password:

After typing in password:

All of the users are on domain machines on the local lan. It works fine externally, probably because everything uses basic auth.

Also, outlook anywhere is set to NTLM for internal connections:

InternalHostname                   : mail.contoso.org
InternalClientAuthenticationMethod : Ntlm
InternalClientsRequireSsl          : True

See the information in the below KB.

Users of Exchange Server 2013 or Exchange Online can't open public folders or shared mailboxes on an Exchange 2010 or Exchange 2007 server
http://support.microsoft.com/kb/2

Found my problem/solution. 

http://support.microsoft.com/kb/2839517/en-us

I have to test it tomorrow. I hope Microsoft comes out with a fix for outlook 2013 soon.