Yes it certainly can however it's not a native feature.
What you could do is add a custom userfield during registration to show the checkbox and if they click yes or no show a message directing them to pay on all pages until such time as they pay. It's not as clean but it would work.
If you wanted to do it smoothly you would need an external hack. I'm not aware of any hacks that do this as you've described it or anything close but it's completely possible.
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