How do I add the "- Any -" option for the published status on exposed filters?
As the screenshot shows, the filter is set as Required. This means the users need to select a value for that filter that, being a filter for the published status, can only get Yes and No as values.

If you want to allow users to select - Any - as value for that filter (which isn't really a value for the published status, but a value that tells the view not to filter the nodes basing on their published status), you need to deselect the checkbox that is shown to the left of Required.