You may want to check out Square Invoices feature for Deposits...

You can set up multiple payments to receive - or you can store the Customer's Card on file and simply run it once every month. 

https://squareup.com/help/us/en/article/6581-request-deposits-with-square-invoices

If this doesn't do what you need exact enough, we can easily make a system that allows your Customers to pay any custom amount on any date you (or they) specify. It'll save their payments for history, store their card securely (using Square's token method) and tie perfectly back to your current Square Account and Dashboard so you'll see everything. 

Note: We make custom websites and mobile apps that are powered by Square. We pre-build and already have a lot of technology so it's easy for us to make stuff like this. 

Ken

https://getpreorder.com/square-pos