Your code does not work because you are not assigning a variable email_addressanywhere. You must change this value to use $('#email_address').val().
, jsFiddle ; URL- cookie script Github. CDN. :
var cookie_email = Cookies.get('user_email');
$('#email_address').val(cookie_email);
$('#test_button').click(function() {
Cookies.set('user_email', $('#email_address').val(), {
expires: 365
});
});