How to update a field on menu_link_content (added using menu_item_extras module) entity from node form
There are couple of problems here: 


The node form has the menu fields hardcoded. See menu_ui_form_node_form_alter() and menu_ui_form_node_form_submit() for more details.
Turns out the root cause of the field issue is that the Menu Item Extras module updates the bundle value to menu name which happens on pre_save hook of menu_link_content entity. Without menu_name as bundle, the entity wouldn't have any fields associated. 


One way to solve this issue is to implement a hook_form_node_form_alter() hook and update the bundle details before trying to update the custom field. Here is the code:

1. mymodule/mymodule.module


/**
 * Implements hook_form_BASE_FORM_ID_alter() for \Drupal\node\NodeForm.
 */
function mymodule_form_node_form_alter(&amp;$form, FormStateInterface $form_state) {
  foreach (array_keys($form['actions']) as $action) {
    if ($action != 'preview' &amp;&amp; isset($form['actions'][$action]['#type']) &amp;&amp; $form['actions'][$action]['#type'] === 'submit') {
      $form['actions'][$action]['#submit'][] = [
        \Drupal::service('mymodule.menu_link_content_helper'),
        'nodeSubmit',
      ];
    }
  }
}



2. mymodule/src/MenuLinkContentService.php

&lt;?php

namespace Drupal\mymodule\Services;

use Drupal\Core\Entity\EntityTypeManagerInterface;
use Drupal\Core\Form\FormStateInterface;
use Drupal\menu_link_content\Entity\MenuLinkContent;

/**
 * Class MenuLinkContentService
 *
 * @package Drupal\mymodule\Service
 */
class MenuLinkContentService  {

  /**
   * Entity type manager.
   *
   * @var \Drupal\Core\Entity\EntityTypeManagerInterface
   */
  protected $entityTypeManager;

  /**
   * MenuLinkContentHelper constructor.
   *
   * @param \Drupal\Core\Entity\EntityTypeManagerInterface $entityTypeManager
   *   Entity type manager.
   */
  public function __construct(EntityTypeManagerInterface $entityTypeManager) {
    $this-&gt;entityTypeManager = $entityTypeManager;
  }

  /**
   * Form submission handler for menu item field on the node form.
   *
   * @param array $form
   *   Form array.
   * @param \Drupal\Core\Form\FormStateInterface $form_state
   *   Form state object.
   *
   * @throws \Drupal\Component\Plugin\Exception\InvalidPluginDefinitionException
   * @throws \Drupal\Component\Plugin\Exception\PluginNotFoundException
   * @throws \Drupal\Core\Entity\EntityStorageException
   */
  public function nodeSubmit(array $form, FormStateInterface $form_state) {
    if (!$form_state-&gt;isValueEmpty('menu')) {
      $values = $form_state-&gt;getValue('menu');
      if (!empty($values['entity_id'])) {
        // There is no bundle by default for menu_link_content entity.
        $menu_name = 'menu_link_content';
        if (!empty($values['menu_parent'])) {
          list($menu_name) = explode(':', $values['menu_parent'], 2);
        }
        $entity = $this-&gt;nodeSaveMenuLinkContent($values['entity_id'], $menu_name);
        // If no  menu item with given bundle, load and update bundle.
        if (!$entity) {
          $entity = MenuLinkContent::load($values['entity_id']);
          $entity-&gt;set('bundle', $menu_name)-&gt;save();
          $this-&gt;nodeSaveMenuLinkContent($entity-&gt;id(), $menu_name);
        }
      }
    }
  }

  /**
   * Updates additional field on menu link content entity.
   *
   * @param int $entity_id
   *   Entity ID.
   * @param string $menu_name
   *   Menu name.
   *
   * @return \Drupal\Core\Entity\EntityInterface|mixed|null
   *   An Entity. NULL, if no entity with given ID and bundle found.
   *
   * @throws \Drupal\Component\Plugin\Exception\InvalidPluginDefinitionException
   * @throws \Drupal\Component\Plugin\Exception\PluginNotFoundException
   * @throws \Drupal\Core\Entity\EntityStorageException
   */
  protected function nodeSaveMenuLinkContent($entity_id, $menu_name) {
    $storage = $this-&gt;entityTypeManager-&gt;getStorage('menu_link_content');
    // Using loadByProperties() as loading without bundle wouldn't bring the
    // custom field.
    $entities = $storage-&gt;loadByProperties([
      'bundle' =&gt; $menu_name,
      'id' =&gt; $entity_id,
    ]);
    $entity = reset($entities);
    if ($entity) {
      $entity-&gt;set('field_searchable', TRUE);
      $entity-&gt;save();
      return $entity;
    }
    else {
      return NULL;
    }
  }

}



Note: my checkbox field is field_searchable.