How to submit 0 if checkbox is unchecked and submit 1 if checkbox is checked in HTML

Simplest one, no javascript required, just put a hidden input before the checkbox:

<input type="hidden" name="check[0]" value="0" />
<input type="checkbox" name="check[0]" value="1" />

Inputs need to have the same name. If the checkbox is checked then value 1 will be submitted, otherwise value 0 from the hidden input.

Your case javascript solution, no hidden inputs needed:

<script type="text/javascript">
    // when page is ready
    $(document).ready(function() {
         // on form submit
        $("#form").on('submit', function() {
            // to each unchecked checkbox
            $(this + 'input[type=checkbox]:not(:checked)').each(function () {
                // set value 0 and check it
                $(this).attr('checked', true).val(0);
            });
        })
    })
</script>

<form method="post" id="form">
    <input type="checkbox" name="check[0]" value="1" />
    <input type="checkbox" name="check[1]" value="1" />
    <input type="submit" value="Save Changes" />
</form>

PHP solution, no hidden inputs needed:

<?php
    // if data is posted, set value to 1, else to 0
    $check_0 = isset($_POST['check'][0]) ? 1 : 0;
    $check_1 = isset($_POST['check'][1]) ? 1 : 0;
?>

<form method="post">
    <input type="checkbox" name="check[0]" value="1" />
    <input type="checkbox" name="check[1]" value="1" />
    <input type="submit" value="Save Changes" />
</form>

EDIT: the javascript solution is not valid anymore as of jquery 1.6. Based on this, a more proper solution is the following:

<script type="text/javascript">
    // when page is ready
    $(document).ready(function() {
         // on form submit
        $("#form").on('submit', function() {
            // to each unchecked checkbox
            $(this).find('input[type=checkbox]:not(:checked)').prop('checked', true).val(0);
        })
    })
</script>

<form method="post" id="form">
    <input type="checkbox" name="check[0]" value="1" />
    <input type="checkbox" name="check[1]" value="1" />
    <input type="submit" value="Save Changes" />
</form>