TL;DR: The short answer is no. Keep on reading if you want a longer explanation.

Can I move my Beeper Android app to an external storage (SD Card)?

Moving certain installed apps from your Android phone's internal storage to external storage can free up some storage space. However, Beeper Android, though it might be possible to move to external storage, is 100% designed to work installed in the device's internal storage. 

Though you might be able to receive new message notifications while the app is installed on external storage, the app is expected to crash upon opening.