If you want to re-instantiate your countdown timer only after the date has been changed and saved, then, something to the effect of:
//In your custom.js file....
//make sure your localized data is a dependency of your custom.js file so you have proper scope to local data
jQuery(document).ready(function($){
var date, gapinvite;
date = $('input[name="director_date"]');
if ( date.val().length ) {
gapinvite = new Date(Data.date); //if localizing variable
/* OR */
//gapinvite = new Date(date.val()); //if accessing variable directly from value attribute
} else {
/* if gapinvite should always start from 2014, 06 - 1, 2 as default do... */
gapinvite = new Date(2014, 06 - 1, 2);
/* if gapinvite should always start from today as default do... */
//gapinvite = new Date(); //uncomment if required
/*
* if gapinvite should always come from your default placeholder value do...
* how you generate that value is up to you.
*/
//gapinvite = new Date(date.attr('placeholder')); //uncomment if required
}
$('.days').countdown({
until: gapinvite,
layout: '{dn} {dl}',
/* Set your timezone */
timezone: gapinvite.getTimezoneOffset()
});
});
This will grab the default placeholder date if the director_date value attribute is empty, otherwise it will grab the value supplied by the user and instantiate the countdown timer accordingly.
In future, questions strictly related to JavaScript and jQuery should be asked at Stackoverflow
Edit: remove unnecessary timezone variable, calculate offset directly within object property value.