You can do it in 3 steps:
- Adding the custom checkbox field below the Payment Methods
 
- Saving the custom checkbox field when it's checked in the order meta
 
- Displaying the custom checkbox field when it's checked on the order edit page
 
Here is that code:
// Add custom checkout field: woocommerce_review_order_before_submit
add_action( 'woocommerce_review_order_before_submit', 'my_custom_checkout_field' );
function my_custom_checkout_field() {
    echo '<div id="my_custom_checkout_field">';
    woocommerce_form_field( 'my_field_name', array(
        'type'      => 'checkbox',
        'class'     => array('input-checkbox'),
        'label'     => __('My custom checkbox'),
    ),  WC()->checkout->get_value( 'my_field_name' ) );
    echo '</div>';
}
// Save the custom checkout field in the order meta, when checkbox has been checked
add_action( 'woocommerce_checkout_update_order_meta', 'custom_checkout_field_update_order_meta', 10, 1 );
function custom_checkout_field_update_order_meta( $order_id ) {
    if ( ! empty( $_POST['my_field_name'] ) )
        update_post_meta( $order_id, 'my_field_name', $_POST['my_field_name'] );
}
// Display the custom field result on the order edit page (backend) when checkbox has been checked
add_action( 'woocommerce_admin_order_data_after_billing_address', 'display_custom_field_on_order_edit_pages', 10, 1 );
function display_custom_field_on_order_edit_pages( $order ){
    $my_field_name = get_post_meta( $order->get_id(), 'my_field_name', true );
    if( $my_field_name == 1 )
        echo '<p><strong>My custom field: </strong> <span style="color:red;">Is enabled</span></p>';
}
Code goes in function.php file of your active child theme (or theme) or also in any plugin file.
Tested in WooCommerce 3+ and works. When the checkbox has been checked, it display a custom text below billing address in order edit page…