You haven't done anything wrong, and it isn't a hidden block. Square Appointments books one-to-one: an appointment makes the team member it's assigned to busy for the whole length of the service, and online customers can't book on top of that.
On your booking page, your boarding packages show as 24-hour services, and they're all assigned to you as the only team member. So once one dog books, Square treats you as busy for those 24 hours. A new 24-hour booking can't overlap it on either side, so one stay can close off close to two days of start times. That's how a whole weekend shows full when you still have room. Your free Meet & Greet is assigned to you too, so it gets blocked the same way.
Square doesn't really do multi-night stays, so the practical fix is to stop it holding your time:
1. Edit each boarding service and set its duration to just the drop-off, say 15 or 30 minutes. Keep the number of nights in the service name and price. Other customers can then book the same days, and Meet & Greets open back up. The trade-off: Square won't cap how many dogs you take, so keep your own count of who's in each night.
2. Then set your online booking hours to your real drop-off times. Right now the page offers a start time every 15 minutes around the clock, 2 AM included.
3. Bookings that come in by phone or text can still be added from your own Square calendar even when that time looks taken. Square only stops customers from overlapping online.
To confirm before you change anything, open one package in your Square dashboard and check its duration. If it says 24 hours, that's the cause.
Hope that gets your weekends open again.
