Looking for a simple approach for handling user $_POST data without AJAX?

The best way to process the custom form is the following. If you are using nonce then you don’t really have to check $_POST[‘checkbox’], the code below can be used simply to verify_nonce and then process the form.

function process_my_form() {
    if ( wp_verify_nonce( $_POST['my_nonce_field'], 'my_nonce' ) ) {
        // process your form here
        // you can also redirect and call exit here.
    }
}
add_action( 'init', 'process_my_form' );

But if you still you want to submit your form to a custom php script, then load the wp-blog-header.php.

<?php 
define('WP_USE_THEMES', false);
require('./wp-blog-header.php');
?>

See http://codex.wordpress.org/Integrating_WordPress_with_Your_Website for more information.