To change that, you'd want to change the URL those links point to, which in this case is tickets.php, however, "My Tickets" is just another version of the same "Open" listing, we really only need to change a single variable!
In include/staff/ticket-view.inc.php
find:
<form action="tickets.php?id=<?php echo $ticket->getId(); ?>" method="post" id="status-form" name="status-form">
Change to:
<form action="tickets.php?status=assigned&id=<?php echo $ticket->getId(); ?>" method="post" id="status-form" name="status-form">
So, we are just adding status=assigned& into the url!
Now requests from those buttons will point to the correct list.
Tested, works.