<!-- Find the following line in your custom events/single.php template -->#
<div class="navlink tribe-previous"><?php tribe_previous_event_link(); ?></div>

<!-- Change it to the following -->
<div class="navlink tribe-previous">
    <?php
    ob_start();
    tribe_previous_event_link();
    $prev_event = trim(ob_get_clean());

    echo $prev_event;
    if (!empty($prev_event)) echo ' | ';
    ?>
</div>