phpmailer connection refused .. line 254

My mail server is accepting mail() connections. However, I'm changing to phpmailer and got an error message that I don't understand.
Here's the phpmailer code that I'm using:

<?php
            $subject = "Registration Confirmation";

            // The non-HTML text
            $alt_body = "\r\n** CONFIDENTIAL **\r\n\r\n";
            $alt_body .= "Thank you for registering.\r\n\r\n";

            // HTML text
            $body = "** CONFIDENTIAL **<br><br>";
            $body .= "Thank you for registering.<br><br>";

            $mail = new PHPMailer;

            $mail->SMTPDebug = 3;                               // Enable verbose debug output   ####################### CHANGE TO 0 FOR PRODUCTION ##############################

            $mail->isSMTP();                                      // Set mailer to use SMTP
            $mail->Host = 'smtp.example.com';  // Specify main and backup SMTP servers
            $mail->SMTPAuth = true;                               // Enable SMTP authentication
            $mail->Username = "support@example.com";                 // SMTP username
            $mail->Password = "password";                           // SMTP password
            $mail->SMTPSecure = 'none';                            // Enable TLS encryption, `ssl` also accepted
            $mail->Port = 587;                                    // TCP port to connect to

            $mail->From = 'support@example.com';
            $mail->FromName = "Support";
            $mail->addAddress("name@gmail.com");     // Add a recipient
            $mail->addReplyTo('support@example.com', 'Support');
            $mail->WordWrap = 50;
            $mail->isHTML(true);                                  // Set email format to HTML

            $mail->Subject = $subject;
            $mail->Body    = $body;
            $mail->AltBody = $alt_body;

            if(!$mail->send()) {
                echo 'Message could not be sent.';
                echo 'Mailer Error: ' . $mail->ErrorInfo;
            } else {
                echo 'Message has been sent';
            }
?>

And the error message:

2015-04-10 18:46:57 Connection: opening to smtp.example.com:587, t=300, opt=array ( )

Error '2'
occurred in script '/folders/includes/class.smtp.php' on line 254:
stream_socket_client(): unable to connect to smtp.example.com:587 (Connection refused)
with 'Array'
Date/Time: 10-4-2015 14:46:57

2015-04-10 18:46:57 SMTP ERROR: Failed to connect to server: Connection refused (111) 2015-04-10 18:46:57 SMTP connect() failed. Message could not be sent.Mailer Error: SMTP connect() failed.temp exit

And the chunk of class.smtp.php code that's pointed at:

<?php
// In class.smtp.php ..
        if ($streamok) {
            $socket_context = stream_context_create($options);
            //Suppress errors; connection failures are handled at a higher level
            $this->smtp_conn = @stream_socket_client(
                $host . ":" . $port,
                $errno,
                $errstr,
                $timeout,
                STREAM_CLIENT_CONNECT,
                $socket_context
            );                                     // <<<<<<<<<<<<< LINE 254 IN class.smtp.php <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

I cannot find where STREAM_CLIENT_CONNECT is defined in either class.smtp.php or class.phpmailer.php . Could that be the issue?
Ideas?