Can I backup or mirror an IMAP mailbox on Mac OS X

You can do it with offlineimap. I'm using it for my GMail and Fastmail accounts.

Install Homebrew (you must also install Xcode). With Homebrew, brew install offline-imap.

Create ~/.offlineimaprc config file:

[general]
ui = ttyui
accounts = fastmail
maxsyncaccounts = 4
socktimeout = 60

[Account fastmail]
localrepository = fastmail-local
remoterepository = fastmail-remote

[Repository fastmail-local]
type = Maildir
localfolders = ~/Mail/Fastmail

[Repository fastmail-remote]
type = IMAP
remotehost = mail.messagingengine.com
remoteuser = username
remotepass = password

expunge = no
folderfilter = lambda foldername: foldername not in ['INBOX.Trash']

A big advantage of offlineimap is that it creates the folders you have on your IMAP server.

I'm reading my backup with mutt.

Try:

function TJ() {
  var threads = GmailApp.search('label:inbox is:starred');
  for (var h = 0; h < threads.length; h++) {
    var messages = threads[h].getMessages();
    for (var i = 0; i < messages.length; i++) {
      if (messages[i].isStarred())
      {
        Logger.log(messages[i].getSubject());
        messages[i].unstar();
        messages[i].forward("recipient1@example.com", {
          // Advanced parameters: https://developers.google.com/apps-script/reference/gmail/gmail-message#forward%28String,Object%29
          cc: "myboss@example.com",
          bcc: "mybosses-boss@example.com,vp@example.com"
          });     
      }
  }
}
}

For more information regarding Google Apps Script, see this post: http://www.johneday.com/422/time-based-gmail-filters-with-google-apps-script