Okay... so it does appear to be a problem with the phone's input (headphone) jack. Someone from techsupport gave me an idea. If you're having trouble try this:

Play music through your phone speaker.

Plug in headphones. 

If the jack is identifying devices properly, the speaker shuts off and the headphones play music. This actually did NOT occur when I plugged in headphones. The speaker kept playing. 

In this case, the problem appears to be with our equipment, not Square's. Really appreciate all the help. We've been wrestling with this for weeks. We'll get the phone problem fixed and try again.

Hey @HypnosisRandy, thanks for circling back. I'm sorry to hear about the headphone jack, but glad you were able to get more clarity with our Support team. Hopefully, you're able to get the phone repaired without too much trouble. If anything else comes up, please let us know! 👍

Have done everything suggested in the troubleshooting a number of times. My device (HTC One M9) works with chip reader and manually.  The app recognizes the existence of the magstripe reader but will not take any input from it.  If I "practice" I get no response, including the predicted message (for an Android device) that I need to charge at least $1.00.  If I practice with a $1.00 charge I still get no response.  I've replaced the magstripe reader but still nothing.

Sorry to hear that, @PJB.

If you've ran through all of our suggested troubleshooting tips, I'm afraid there isn't much more I can suggest. 

The HTC One M9 isn't currently listed among our supported devices, but that doesn't necessarily mean one or all of our Reader will not work with it - we just don't have enough data to officially support it. 

Apologies again for the trouble, keep an eye out for future updates.

Sorry to hear about the trouble, @ultrublu. Have you had a chance to run through these troubleshooting tips yet? Generally, those take care of any swiping issues with the magstripe reader. Let me know!

Thanks for responding. I tried all the tips but no luck.  I don't think there's a problem with the reader because it works fine on my IPAD. And I don't think it's my phone because the reader does not work on my wife's Samsung either. On both phones, when I swipe, I get "Reader connected. Swipe to charge." Swipe again and I get "Magstripe reader disconnected."

And you do not have a case on your phone? Is your wife's phone the same model?