Here is a jsFiddle:
http://jsfiddle.net/ezanker/LS3g6/1/The HTML markup to include a hidden datebox in the header with a button at top right is as follows:
<div data-role="header">
<h1>My page</h1>
<a data-role="button" id="myplus" class="ui-btn-right" >+</a>
<input name="mydate" id="mydate" type="date" data-role="datebox" data-options='{"mode": "calbox", "hideInput": "true", "centerHoriz": "true", "closeCallback":"onCLoseMyDate();"}' />
</div>
Setting hideInput to true hides the calendar input box, centerHoriz centers the popup on the screen and the closeCallback is called when you select a date and close the popup.
To launch the popup, handle the click event on the myplus button:
$('#myplus').on('click', function() {
$('#mydate').datebox('open');
});
Then supply the callback function:
function onCLoseMyDate(){
var dateObject = $('#mydate').datebox('getTheDate');
alert(dateObject);
}