Using Conditional Tags to restrict something to 1 user?

I don’t know the plugin, but with a rapid eye on it, it seems that what you want is not, but you can rely on widget_content filter fired by the plugin you are using.

So, in your theme function.php put:

if ( function_exists('widget_logic_redirected_callback') )
   add_filter('widget_content', 'my_widget_for_user', 10, 2);

function my_widget_for_users ($widget_content, $widget_id) {
   $allowed_users = array('username1', 'username2');
   $restricted_widgets = array('widget_id_1', 'widget_id_2');
   global $current_user;
   get_currentuserinfo();
   if ( in_array($widget_id, $restricted_widgets) && ( empty($current_user->user_login) || ! in_array($current_user->user_login, $allowed_users) ) ) {
      $widget_content="";
   }
   return $widget_content;
}

Doing so, if there is no user logged or the user is not an allowed one the widget is technically displayed but with no content so you will see… nothing.