$('#chkBox').click(function() {
    // I prefer assigning a string
    $('#hiddenBox').val(this.checked ? 'true' : 'false');

    // but in actual fact, this should be enough
    $('#hiddenBox').val(this.checked);
});

Demo: http://jsfiddle.net/pGkGz/

$('#chkBox').click(function() {
    // I prefer assigning a string
    $('#hiddenBox').val(this.checked ? 'true' : 'false');

    // but in actual fact, this should be enough
    $('#hiddenBox').val(this.checked);
}).triggerHandler('click');​​​

Demo: http://jsfiddle.net/pGkGz/1/

And see http://api.jquery.com/triggerHandler/