Can I charge my subscribers on a specific day?

Yes, you can.
In order to charge your customers on a specific day of the week/month, you must have an auto-charging subscription rule with a selling plan with a weekly or monthly interval type. When the weekly or monthly interval type is selected, you will see an option to select a day in the week or month when you want to charge your customers.

Here is an example:
Let's say that you have a monthly subscription and you configured the app to charge your customers on 5th day of the month. Your customer creates a subscription on 2021-07-01.
The next charge would normally be scheduled for 2021-08-01. However, because you set the app to charge subscribers on 5th day of the month, the system will charge customers, which would normally be charged in the period from 2021-07-05 to 2021-08-05, on 2021-07-05. In this case, your customer will get charged on 2021-07-05.
When using this functionality, it is also recommended to set up the number of cutoff days to a number that will suit your needs.
In our previous example, if you set the number of cutoff days to 7, then all orders that were made more than 7 days before the 5th of the month will get scheduled on the first 5th of month that comes after the initial order. But any order created in the period of 7 days before the 5th of the month will automatically skip the next 5th day and will be scheduled in the next month. For example, if you set it the number of cutoff days to 7 and the customer places an order on 30th of June, their next order will get scheduled for 5th of August. But if they place an order on 27th of June, then the next charge will get scheduled for 5th of July.