How to have a checkbox uncheck another checkbox?

If the data entry will be on forms, you can use Form Rules (assuming that you are on at least the Premier (middle) Quick Base plan and not Essentials.

Set up a rule
Condition:
When Updates needed is checked

Action:
Change the value in Submitting a record for approval to Unchecked.

(and uncheck that box at the bottom of the form for only fire when this condition becomes true)

here is some help on form rules
https://help.quickbase.com/user-assistance/form_rules.html

Would it be more helpful to have those 'clicks' be buttons, rather than checkboxes?  So you can use them in reports, dashboards, and in view mode on a form?

If so I'd suggest creating 3 formula-url buttons that do the editing action of checking or unchecking as needed.