CSS file not registering through add_action

The primary issue is an incorrect template conditional tag:

if( is_page( 'home' ) ) {}

This returns true if the current context is a static page, with the slug home. I presume that you’re actually wanting to test for the Site Front Page – in which case you need to use the is_front_page() conditional:

if ( is_front_page() ) {
    wp_enqueue_style( 'flexslider' );
}

If the stylesheet still isn’t being output, we’ll need further debugging information.