I really can't believe something so simple is causing such a headache.
<input type='checkbox' name="phoneConsent" id="phoneConsent" value="1">
Whether or not this checkbox is checked, it still posts the value "1" for phoneConsent -- shouldn't it not be posting anything?
I think this is the problem:
// variables for data
$(this).find('[name]').each(function(index, value) {
    var that = $(this),
        name = that.attr('name'),
        value = that.val();
        // load loaded variables into array
        form_data[name] = value;
    });
how can I specify that checkboxes that are not set shouldn't have their values posted?
php:
        $phoneConsent = (isset($_POST['phoneConsent'])) ? 1 : 0;
        // Set up the data that is headed to the table
        $data = Array(               
            'phoneConsent'   => $phoneConsent
        );
the actual HTML
<div class = "form-group">
                <label for="phoneConsent" class='checkbox-inline <?php echo ($this_user[0]['subscription'] == 1 ? '' : 'disabledClass') ;?>'>
                    <input type='checkbox' name="phoneConsent" id="phoneConsent" value="1" <?php echo ($this_user[0]['phoneConsent'] == 1 ? ' checked ' : '') ;?> <?php echo ($this_user[0]['subscription'] == 1 ? '' : ' disabled ') ;?>>
                    Premium account holders: check this box to post your phone number to your teaching profile.
                </label>
            </div>
entire HTML form
<form method="post" class = "multi-form teacher_account_form" id="personal_form" name="personal_form">
            <div class ="form-group">
                <label for="country">What country are you from?</label>
                <!-- for selecting the country from db -->
                <div id = "countryfromdb" class = "hidden"><?=$this_user[0]['country_prefix'];?></div>
                <?php $form->select("country", "country_list") ;?>
            </div>
            <div class = "form-group">
                <label for="chinese_name">What is your Chinese name?</label>
                <input type = "text" class="form-control" name="chinese_name" id="chinese_name" value="<?=$this_user[0]['chinese_name']?>">
            </div>
            <div class = "form-group">
                <label for="phone">What is your phone number?*</label>
                <input type="text" class="form-control bfh-phone" data-format="+86 ddd dddd dddd" name="phone" id="phone" value="<?=$this_user[0]['phone']?>">
            </div>
            <div class = "form-group">
                <label for="phoneConsent" class='checkbox-inline <?php echo ($this_user[0]['subscription'] == 1 ? '' : 'disabledClass') ;?>'>
                    <input type='checkbox' name="phoneConsent" id="phoneConsent" <?php echo ($this_user[0]['phoneConsent'] == 1 ? ' checked ' : '') ;?> <?php echo ($this_user[0]['subscription'] == 1 ? '' : ' disabled ') ;?>>
                    Premium account holders: check this box to post your phone number to your teaching profile.
                </label>
            </div>
            <div class="form-group">
                <button type="submit" class="btn btn-default btn-med account_submit" id="personal_submit" name="personal_submit">Update Personal</button>
            </div>
            <small style="color:red;">*Your phone number will NOT appear on your public profile without your permission.</small>
        </form>
jQuery
  var which_button;
//$('.account_submit').click(function() {
$('form.teacher_account_form').on('submit', function(e) {
    e.preventDefault();
    which_button = $(this).attr('id');
    // Create empty jQuery objects -
    var $jsOutput = $([]);
    var $jsForm = $([]);
    // url to submit to
    var ajaxURL = "/users/p_teacher_account_work";
    // Assign jQuery selector objects
    switch (which_button) {
        case "pay_form":
            $jsOutput = $('#pay_output');
            $jsForm = $('#pay_form');
            break;
        case "edu_form":
            $jsOutput = $('#edu_output');
            $jsForm = $('#edu_form');
            break;
        case "image_form":
            $jsOutput = $('#image_output');
            $jsForm = $('#image_form');
            break;
        case "personal_form":
            $jsOutput = $('#personal_output');
            $jsForm = $('#personal_form');
            break;
    }
    // empty data object
    var form_data = {};
    // variables for data
    $(this).find('[name]').each(function(index, value) {
        var that = $(this),
            name = that.attr('name'),
            value = that.val();
        // load loaded variables into array
        form_data[name] = value;
    });
    $.ajax({
        type: 'post',
        url: ajaxURL,
        data: form_data,
        beforeSend: function() {
            //Display a loading message while waiting for the ajax call to complete
            $jsOutput.html("Updating...");
        },
        success: function(response) {
            $jsOutput.html(response);
        }
    });
    return false;
});
console.log form output
checkbox checked
 Object {country: "AI", chinese_name: "", phone: "+86 1", phoneConsent: "on", personal_submit: ""} teacher_account.js:130
checkbox unchecked
Object {country: "AI", chinese_name: "", phone: "+86 1", phoneConsent: "on", personal_submit: ""} teacher_account.js:130
Code finally works. I am very grateful. UPDATED jQuery:
$(this).find('[name]').each(function(index, value) {
        var that = $(this);
        if(that.is(':checkbox'))
        {
            if(that.is(':checked'))
            {
                var name = that.attr('name');
            }
        }
        else
        {
            name = that.attr('name');
        }
            var value = that.val();
        // load loaded variables into array
        form_data[name] = value;
    });
 
     
     
    