If I understand the question, you want to change the value of the checkbox depending if it is checked or not.
Here is one solution:
$('#checkbox-value').text($('#checkbox1').val());
$("#checkbox1").on('change', function() {
if ($(this).is(':checked')) {
$(this).attr('value', 'true');
} else {
$(this).attr('value', 'false');
}
$('#checkbox-value').text($('#checkbox1').val());
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<input type="checkbox" name="acceptRules" class="inline checkbox" id="checkbox1" value="false">
<div id="checkbox-value"></div>