URI baseUri = URI.create("https://my.company.com/");
String username = "";
String password = "";

Authenticator.setDefault(new Authenticator() {

    @Override
    public PasswordAuthentication getPasswordAuthentication() {
        // only return our credentials for our URI
        if (baseUri.getHost().equals(getRequestingHost())) {
            return new PasswordAuthentication(username, password.toCharArray());
        }
        return null;
    }
});

webEngine.setUserAgent("foonAuthorization: Basic YourBase64EncodedCredentials");