$user = R::findOne('users', 'login = ?', array($data['login']));
if( $user )
{
    if ( md5($data['password']) == $user->password) {
        $_SESSION['logged_user'] = $user;
}

<?php
 if( !isset($_SESSION['logged_user']) ) : ?>
   <div class="login">
        <a href="../pages/login.php">Войти</a>
        <a href="../pages/signup.php">Регистрация</a>
    </div>
<?php else: ?>
    <?php
    echo $_SESSION['logged_user']->login; ?> <!--Вот это имя не выводится-->
Авторизован!
<div class="login">
 <a href="../pages/logout.php">Выйти</a>
   </div>
  <?php endif; ?>

object(__PHP_Incomplete_Class)#1 (11) { ["__PHP_Incomplete_Class_Name"]=> string(19) "RedBeanPHPOODBBean" ["properties":protected]=> array(4) { ["id"]=> string(1) "6" ["login"]=> string(5) "ivan " ["email"]=> string(10) "fwefw@fdsf" ["password"]=> string(32) "698d51a19d8a121ce581499d7b701668" } ["__info":protected]=> array(7) { ["type"]=> string(5) "users" ["sys.id"]=> string(2) "id" ["sys.orig"]=> array(4) { ["id"]=> string(1) "6" ["login"]=> string(5) "ivan " ["email"]=> string(10) "fwefw@fdsf" ["password"]=> string(32) "698d51a19d8a121ce581499d7b701668" } ["tainted"]=> bool(false) ["changed"]=> bool(false) ["changelist"]=> array(0) { } ["model"]=> NULL } ["beanHelper":protected]=> object(__PHP_Incomplete_Class)#2 (1) { ["__PHP_Incomplete_Class_Name"]=> string(44) "RedBeanPHPBeanHelperSimpleFacadeBeanHelper" } ["fetchType":protected]=> NULL ["withSql":protected]=> string(0) "" ["withParams":protected]=> array(0) { } ["aliasName":protected]=> NULL ["via":protected]=> NULL ["noLoad":protected]=> bool(false) ["all":protected]=> bool(false) }

var $_user = new StdClass();
$_user->login = $user->login;
...

$_SESSION['logged_user']['login'] = $user->login;
...