Change default status to blank?

The short answer: the next dev will have a new Advanced setting, Hide blank and empty statuses above textfield, which will do what I think you're asking. Run update.php immediately after upgrading the module.

The long answer:

The default status is blank. "does not have a status" is just placeholder text that appears on user profiles and above the status update form when no status has been entered yet or the status is blank, but it's not the actual status. You can make that line go away completely by checking Hide status above textfield at admin/settings/facebook_status/advanced or you could use a translation system (typically the core localization module and/or i18n) to just hide that text specifically.

There are several other more complex solutions, and you would have to refresh your system cache by running update.php before you would see the effects:

You could override the default theme_facebook_status_item() or implement hook_facebook_status_render() in a custom module, or even write a new input filter to run over the status and filter out just the unwanted text. Removing "does not have a status" from the module file itself is considered bad practice because it means you have to do that every time you upgrade the module, but that would work if you refreshed your cache afterwards.

Automatically closed -- issue fixed for 2 weeks with no activity.