I would just make it: value="1" in the html and then use:
$UpdateQuery->bindColumn("ProductNewArrival", "c", "".((isset($_POST["newarrival"]))?$_POST["newarrival"]:"0") ."", "WA_DEFAULT");
If you tell it to use the default when it is blank then it will make the value whatever the default in the database is when unchecked. It would make sense to maybe ignore that last value for checkboxes and always set to zero or one but this is a good workaround.