Super easy, I think:
Place conditional formatting on your submit button (assuming a custom, non-code submit button on the canvas that triggers the submit to db) that has this expression:
concat(my:Box1, my:Box2, my:Box3, my:Box4, my:Box5, my:Box6, my:Box7) = "falsefalsefalsefalsefalsefalsefalse"
You basically create a condition that says, "The expression: concat(my:Box1, my:Box2, my:Box3, my:Box4, my:Box5, my:Box6, my:Box7) = "falsefalsefalsefalsefalsefalsefalse", then disable this control.
This assumes your boolean fields default to "false" when not checked. If it's something else like FALSE, False, or No, then substitute the 7 falses with 7 of whatever you need. The concat is adding each box's value together in a string, and when all unchecked ,the string should be false 7 times. Any time a box is checked, that value changes to true, and the button will become enabled. If the user checks a box and then unchecks it, the condition is triggered again.