TL;DR: If you find yourself in this situation, using Beeper Android, the walk-around is to connect to Google Chat on Beeper Desktop.

Are you unable to connect to Google messages or Google chat with "Couldn't sign you in: This browser or app may not be secure?"?

If you are unable to connect to Google Messages or Google Chat with an Unsupported Browser error, “Couldn't sign you in: This browser or app may not be secure,” the walk-around is to use Beeper Desktop to connect your Google Chat network.