Does the checkbox submit as ‘checked’ ok, but just not when unchecked?
If so, the problem is likely due to the fact that by default, a checkbox does not submit anything to the server if it is not checked (so it can’t be submitted as unchecked).
This issue, and a possible solution, is discussed here:
BUG: Boolean Checkbox in Customer Submitted Item
Hope this can work for you.
If the checkbox value is not being submitted even when ‘checked’ there might be another issue.
In this case, could you post the form and JS code and/or a link to the page?