My experience is that, once configured, the Outlook client on an Internet Facing Deployment prompts for login on launch. I just configured a user and that doesn't happen. For her to get in she must first browse to the CRM, check save password during login, the while the web client is open start Outlook.
Until she reboots, Outlook is logged. Once she reboots these steps are required.
Why / what setting is keeping Outlook for prompting for login on its own?