I am trying to hide a popup when the user clicks anywhere on the screen other
than the popup. My first approach was to add the MouseEvent.CLICK to the
stage, but the child UIComponents on the screen were blocking the click event
from being dispatched from the stage. Is there a way to capture the click
event other than adding event listeners to all the children UI components or
making them MouseEnabled=false?

When you say popup, do you mean an mx.controls.Alert, or your own popup?