How jQuery Mobile handles page changes
To understand this situation you need to understand how jQuery Mobile works. It uses ajax to load other pages.
First page is loaded normally. Its HEAD and BODY is loaded into the DOM, and they are there to await other content. When second page is loaded, only its BODY content is loaded into the DOM.
That's why your button is show successfully but click event is not working. Same click event whose parent HEAD was disregarded during the page transition.
If you want to find out more read my other article: Why I have to put all the script to index.html in jquery mobile
Here's an official documentation: http://jquerymobile.com/demos/1.2.0/docs/pages/page-links.html
Unfortunately you are not going to find this described in their documentation. Ether they think this is a common knowledge or they forgot to describe this like my other topics. (jQuery Mobile documentation is big but lacking many things).
Solution 1
In your second page, and every other page, move your SCRIPT tag into the BODY content, like this:
<body>
<script>
// Your javascript will go here
</script>
// And rest of your HTML content
</body>
This is a quick solution but still an ugly one.
Solution 2
Move all of your javascript into the original first HTML. Collect everything and put it inside a single js file, into a HEAD. Initialize it after jQuery Mobile has been loaded.
<head>
<meta name="viewport" content="width=device-width; initial-scale=1.0; maximum-scale=1.0; minimum-scale=1.0; user-scalable=no; target-densityDpi=device-dpi"/>
<link rel="stylesheet" href="http://code.jquery.com/mobile/1.2.0/jquery.mobile-1.2.0.min.css" />
<script src="http://code.jquery.com/mobile/1.2.0/jquery.mobile-1.2.0.min.js"></script>
<script src="index.js"></script> // Put your code into a new file
</head>
In the end I will describe why this is a part of a good solution.
Solution 3
Use rel="external" in your buttons and every elements you are using to change page. Because of it ajax want be used to load pages and your jQuery Mobile app will behave like a normal web application. Unfortunately this is not a good solution in your case. Phonegap should never work as a normal web app.
<a href="#second" class="ui-btn-right" rel="external">Next</a>
Official documentation, look for a chapter: Linking without Ajax
Realistic solution
Realistic solution would use Solution 2. But unlike solution 2, I would use that same index.js file and initialize it inside a HEAD of every possible other page.
Now you can ask me WHY?
Phonegap like jQuery Mobile is buggy, and sooner or later there's going to be an error and your app will fail (including loaded DOM) if your every js content is inside a single HTML file. DOM could be erased and Phonegap will refresh your current page. If that page don't have javascript that it will not work until it is restarted.