Adding Additional Variables on Menus Page

Use the checked function.
https://codex.wordpress.org/Function_Reference/checked

The HTML you want to render should look like this:

<input ... value="hide" checked="checked" />Hide from these locations.</br>
<input ... value="show" />Only show to these locations.</br>

What you’re getting instead is probably somthing like this:

<input ... value="hide" />Hide from these locations.</br>
<input ... value="" />Only show to these locations.</br>

This should help:

<input type="radio" id="edit-menu-item-visibility-<?php echo $item_id;>" name="menu-item-show-hide[<?php echo $item_id; ?>]" value="hide" <?php checked( get_post_meta( $item_id, 'hide_show', true ), 'hide', true ); ?>" />Hide from these locations.</br>
<input type="radio"id="edit-menu-item-visibility-<?php echo $item_id; ?>" name="menu-item-show-hide[<?php echo $item_id; ?>]" value="show" <?php checked( get_post_meta( $item_id, 'hide_show', true ), 'show', true ); ?>" />

Only show to these locations.

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