Self-portrait of Bekily the Lemur
http://commons.wikimedia.org/wiki/File:Lemur_self-portrait.jpg

When you run this cmdlet:

New-MailboxExportRequest <username> -FilePath \\servername\sharename\username.pst

…and you get this error message:

Couldn’t connect to the source mailbox.
+ CategoryInfo          : NotSpecified: (0:Int32) [New-MailboxExportRequest], RemotePermanentException
+ FullyQualifiedErrorId : B1C4188,Microsoft.Exchange.Management.RecipientTasks.NewMailboxExportRequest

…the most common problem is that the source mailbox has been hidden from the GAL. Oddly, it doesn’t happen with every mailbox that is hidden. It is probably a combination of factors, such as hidden + litigation hold.

Unhide the mailbox and try the export again.