Exclude Posts From Specific Category from Next and Previous post links

Use the 4th parameter $excluded_terms
http://codex.wordpress.org/Function_Reference/next_post_link

<?php next_post_link( $format, $next, $in_same_term = true, $excluded_terms="402", $taxonomy = 'category' ); ?>

Same for the previous_post_link
http://codex.wordpress.org/Template_Tags/previous_post_link

$previous="<span class="meta-nav">Previous Post</span>";
$next="<span class="meta-nav">Next Post</span>";

<?php previous_post_link( $format, $previous, $in_same_term = true, $excluded_terms="402", $taxonomy = 'category' ); ?>

Or you could create a template tag like whats included in Twenty Fourteen

if ( ! function_exists( 'twentyfourteen_post_nav' ) ) :
/**
 * Display navigation to next/previous post when applicable.
 *
 * @since Twenty Fourteen 1.0
 */
function twentyfourteen_post_nav() {
    // Don't print empty markup if there's nowhere to navigate.
    $previous = ( is_attachment() ) ? get_post( get_post()->post_parent ) : get_adjacent_post( false, '', true );
    $next     = get_adjacent_post( false, '', false );

    if ( ! $next && ! $previous ) {
        return;
    }

    ?>
    <nav class="navigation post-navigation" role="navigation">
        <h1 class="screen-reader-text"><?php _e( 'Post navigation', 'twentyfourteen' ); ?></h1>
        <div class="nav-links">
            <?php
            if ( is_attachment() ) :
                previous_post_link( '%link', __( '<span class="meta-nav">Published In</span>%title', 'twentyfourteen' ) );
            else :
                previous_post_link( '%link', __( '<span class="meta-nav">Previous Post</span>', 'twentyfourteen' ) );
                next_post_link( '%link', __( '<span class="meta-nav">Next Post</span>', 'twentyfourteen' ) );
            endif;
            ?>
        </div><!-- .nav-links -->
    </nav><!-- .navigation -->
    <?php
}
endif;