Send checkbox status with Ajax / JSON and save it

.prop()

method return boolean type. True or false. You should send checked value and save it into database. And in HTML get your value and set it to your checkbox.
An example you have <input type="checkbox" name="test" class="checkbox" value="checked">. You should send to php $('.checkbox').val(); and then place checked attr to html.

<?php $checked = 'your var from database'; ?>
<input type="checkbox" <?php checked('checked', $checked); ?> name="test" class="checkbox" value="checked">

Something like this should work.