As you're creating your buttons dynamically you're interested in the .on()
method:
$('.page').on("click", ".link, .back", function( e ){
e.preventDefault();
var HTML = {
"link" : '<a href="#">New Link</a><button class="back">Go back to First Link</button>',
"back" : '<a href="#" class="link">First Link</a>'
}
$('.page').html(HTML[this.className]);
});
also: restore the class .link
to the newly generated anchor:
<a href="#" class="link">First Link</a>