@rlevine Since you purchased a second reader and it does exactly the same thing, I would suspect the Square app, phone/tablet, or payment settings rather than the reader itself.
First go to More → Settings → Hardware → Square Readers and verify the reader actually says Ready, not simply connected. Square says a properly authenticated reader should show Ready.
Also make sure you have a good internet connection. The Square app needs internet access to authenticate the reader and check/install reader firmware updates.
Since you've already reset, updated, re-paired and even replaced the reader, I wouldn't keep factory-resetting it. I would try signing into your Square account on another compatible phone or tablet, pair the new reader, and run a small test transaction. If tap works there, you've isolated the problem to the original device/app configuration.
If it fails on the second device as well, I'd contact Square Support at 1-855-700-6000 because at that point I'd want Square to check the account/payment configuration rather than replace another reader.
“When you press Charge, does the screen say ‘Tap, Insert, or Swipe,’ and does inserting a chip card still work?”
That's important. Square says that once you press Charge, a correctly functioning Reader should be ready for tap or dip payments. If chip works but tap never activates on two separate readers, that's a much more specific problem than the reader simply losing its Bluetooth connection.
