I received a reply from BigCommerce and it was no help.
I figured this out with the help of "how to use basic authorization in php curl", "JSON to PHP Using json_decode", and "JSON".
I ran this code, in a .php file, on a GoDaddy hosted server running PHP 5.6.22 and curl 7.36.0.
<?php
$username = "Username";
$password = "API Token";
$URL = "https://store.com/api/v2/orders/count.json";
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, $URL);
curl_setopt($ch, CURLOPT_TIMEOUT, 30); //Timeout after 30 seconds
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_HTTPAUTH, CURLAUTH_ANY);
curl_setopt($ch, CURLOPT_USERPWD, "$username:$password");
$status_code = curl_getinfo($ch, CURLINFO_HTTP_CODE); //Get status code
$response = curl_exec($ch);
curl_close($ch);
$result = json_decode($response);
echo "Count = " . $result->count; //Total orders
echo "<br />" . $result->statuses[8]->name . " = " . $result->statuses[8]->count; //Shipped orders
?>