public static RestResponse SendSimpleMessage() {
RestClient client = new RestClient();
client.BaseUrl = "https://api.mailgun.net/v3";
client.Authenticator = new HttpBasicAuthenticator(
"api","key-3ax6xnjp29jd6fds4gc373sgvjxteol0");
RestRequest request = new RestRequest();
request.AddParameter("domain",
"samples.mailgun.org", ParameterType.UrlSegment);
request.Resource = "{domain}/messages";
request.AddParameter("from", "Excited User <excited@samples.mailgun.org>");
request.AddParameter("to", "devs@mailgun.net");

con eso, en algun método, le pones esto:
// Compose a message
MailMessage mail = new MailMessage("foo@YOUR_DOMAIN_NAME", "bar@example.com");
mail.Subject = "Hello";
mail.Body = "Testing some Mailgun awesomness";

// Send it!
SmtpClient client = new SmtpClient();
client.Port = 587;
client.DeliveryMethod = SmtpDeliveryMethod.Network;
client.UseDefaultCredentials = false;
client.Credentials = new System.Net.NetworkCredential("postmaster@YOUR_DOMAIN_NAME", "3kh9umujora5");
client.Host = "smtp.mailgun.org";

private void btnEnviar_Click(object sender, EventArgs e)
{
MailMessage _Correo = new MailMessage();

if (string.IsNullOrEmpty(txtDe.Text))
MessageBox.Show("Remitente no puede estar vacío", "Aviso", MessageBoxButtons.OK, MessageBoxIcon.Information);
else
{
_Correo.From = new MailAddress(txtDe.Text);

_Correo.To.Add(txtPara.Text);
_Correo.Subject = txtAsunto.Text;
_Correo.Body = txtContenido.Text;
_Correo.IsBodyHtml = false; // Le indicamos que el cuerpo del mensaje no es HTLM
_Correo.Priority = MailPriority.Normal;