I realize it's considered "old school" to print-out anything nowadays, but I still print-out all my flight confirmations, boarding passes, hotel and rental car confirmations, etc. Too risky to not get a cell phone signal, internet connection, phone running out of power, etc. I've had hotels try to charge me more than booked and luckily I had the confirmation showing the lower rate. I've had post-flight airline missing miles and they always want a copy of the boarding pass (which usually disappears off the phone after a few days.) If I were you, I'd definitely print the ETA confirmation.