wp_nav_menu() | Outputting my own custom code using built in features

The construction of the nav menu links is controlled by the start_el() method of the Walker_Nav_Menu class. It’s possible to change its behaviour by extending Walker_Nav_Menu to a new class, and replacing the start_el() method with your own. You can then use your new walker by setting the walker argument of wp_nav_menu().

So the first step would be to create a new class (let’s call it Box_Link_Walker), and the copy the entirety of start_el() method of the Walker_Nav_Menu into it (I have removed all the inline documentation of the original method to save space in this answer):

class Box_Link_Walker extends Walker_Nav_Menu {
    public function start_el( &$output, $item, $depth = 0, $args = array(), $id = 0 ) {
        if ( isset( $args->item_spacing ) && 'discard' === $args->item_spacing ) {
            $t="";
            $n = '';
        } else {
            $t = "\t";
            $n = "\n";
        }
        $indent = ( $depth ) ? str_repeat( $t, $depth ) : '';

        $classes = empty( $item->classes ) ? array() : (array) $item->classes;
        $classes[] = 'menu-item-' . $item->ID;

        $args = apply_filters( 'nav_menu_item_args', $args, $item, $depth );

        $class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item, $args, $depth ) );
        $class_names = $class_names ? ' class="' . esc_attr( $class_names ) . '"' : '';

        $id = apply_filters( 'nav_menu_item_id', 'menu-item-'. $item->ID, $item, $args, $depth );
        $id = $id ? ' id="' . esc_attr( $id ) . '"' : '';

        $output .= $indent . '<li' . $id . $class_names .'>';

        $atts = array();
        $atts['title']  = ! empty( $item->attr_title ) ? $item->attr_title : '';
        $atts['target'] = ! empty( $item->target )     ? $item->target     : '';
        $atts['rel']    = ! empty( $item->xfn )        ? $item->xfn        : '';
        $atts['href']   = ! empty( $item->url )        ? $item->url        : '';

        $atts = apply_filters( 'nav_menu_link_attributes', $atts, $item, $args, $depth );

        $attributes="";
        foreach ( $atts as $attr => $value ) {
            if ( ! empty( $value ) ) {
                $value = ( 'href' === $attr ) ? esc_url( $value ) : esc_attr( $value );
                $attributes .= ' ' . $attr . '="' . $value . '"';
            }
        }

        $title = apply_filters( 'the_title', $item->title, $item->ID );

        $title = apply_filters( 'nav_menu_item_title', $title, $item, $args, $depth );

        $item_output = $args->before;
        $item_output .= '<a'. $attributes .'>';
        $item_output .= $args->link_before . $title . $args->link_after;
        $item_output .= '</a>';
        $item_output .= $args->after;

        $output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
    }
}

The only bits we need to worry about are this section:

$atts['title']  = ! empty( $item->attr_title ) ? $item->attr_title : '';
$atts['target'] = ! empty( $item->target )     ? $item->target     : '';
$atts['rel']    = ! empty( $item->xfn )        ? $item->xfn        : '';
$atts['href']   = ! empty( $item->url )        ? $item->url        : '';

Where you just need to add a line that will add the box-link class to the anchor element:

$atts['class']  = 'box-link';

And this section:

$item_output = $args->before;
$item_output .= '<a'. $attributes .'>';
$item_output .= $args->link_before . $title . $args->link_after;
$item_output .= '</a>';
$item_output .= $args->after;

Which just needs to be rearranged to move the title outside of the anchor element:

$item_output = $args->before;
$item_output .= '<a'. $attributes .'>';
$item_output .= '</a>';
$item_output .= $args->link_before . $title . $args->link_after;
$item_output .= $args->after;

Now you can apply this new markup to your menu by setting the walker argument of wp_nav_menu():

wp_nav_menu( array(
    'theme_location' => 'primary-menu',
    'menu_id'        => 'primary-menu',
    'walker'         => new Box_Link_Walker(),
) );

That out of the way, I feel I should point out that structuring your links this way is bad for accessibility, as to screen readers – and search engine robots, for that matter – the link won’t appear to have a label. I would advise against implementing this. Whatever CSS effect you’re trying to apply is almost certainly possible without breaking the links’ accessibility.