You did not add # before id of the button. You do not have right selector in your jquery code. So jquery is never execute in your button click. its submitted your form directly not passing any ajax request.
See documentation: http://api.jquery.com/category/selectors/
its your friend.
Try this:
It seems that id: $("#Shareitem").val() is wrong if you want to pass the value of
<input type="hidden" name="id" value="" id="id">
you need to change this line:
id: $("#Shareitem").val()
by
id: $("#id").val()
All together:
<script src="http://code.jquery.com/jquery-1.11.0.min.js"></script>
<script>
$(document).ready(function(){
$("#Shareitem").click(function(e){
e.preventDefault();
$.ajax({type: "POST",
url: "/imball-reagens/public/shareitem",
data: { id: $("#Shareitem").val(), access_token: $("#access_token").val() },
success:function(result){
$("#sharelink").html(result);
}});
});
});
</script>