The android BA app allows screenshots. I take a screenshot everytime just in case there is a problem with app loading when I'm at the boarding gate (although it has never happened), and also in case I need it as hasn't credited automatically to my Qantas account (which has happened often).
Sometimes the boarding pass sticks around in the app for a while after the flight, I'm not sure for how long though. I assume you have already checked the my boarding passes section on the app.
When checking in on the android BA app instead of choosing the download the boarding pass choose the option to email the boarding pass and you will receive a boarding pass in A4 PDF format which can printed out (or simply kept for a Qantas claim). I don't think this version is intended to be viewed or used on a phone. After having got the boarding pass emailed you can still go back into the android BA app and download a mobile boarding pass to the app.
