With Windows Authentication the Windows Identity the process is running as is what everything is accessed as. What I mean by that is, when your Windows User has access to a UNC path the reason why you are not prompted for a username and password when you go to that path is because Windows knows who you are logged in as and it automatically and transparently passes a token to authenticate you to the external resource. The same happens in unattended processes, e.g. with SSIS Packages running on the server. The service account running the SSIS Package needs access to the UNC path.

Re: Connection Strings, those are for databases only you can only specify a username and password for database-based Logins, not Windows Users. If you have Trusted Authentication=true or Integrated Security=true in your connection string then you would not supply a username and password.