My understanding is that, when clients install my app then the billing gets handled by Square Pos. Is that the case or not?
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No, that’s not the case. The Square Point of Sale (POS) system handles payment processing for transactions, but it does not handle billing for your app. The billing for your app, including any download or subscription fees, would be handled through the platform where your app is available for download (like the App Store or Google Play Store). Square POS only handles the transactions that occur within your app or in-person via Square hardware.
This answer was reviewed by @Bryan-Square.
Did you setup App Subscriptions? 
Unfortunately, it doesn’t seem that the subscription option is available for us here in Canada. what is the alternative?
Oh yeah, it’s currently only available in the US. You can use our Subscriptions API to create and manage subscription.