Jasné. Už je pozdě, prdím na to.

  HttpWebRequest req = (HttpWebRequest)WebRequest.Create("http://www.dovera.sk/overenia/overit-poistenca");
            req.UserAgent = "Mozilla/5.0 (Windows NT 6.1; WOW64; rv:38.0) Gecko/20100101 Firefox/38.0";
            req.Timeout = 5000;
            string postData = "rodne_cislo=861020/1234&submit=submit";
            byte[] send = Encoding.Default.GetBytes(postData);
            req.Method = "POST";
            req.ContentType = "application/x-www-form-urlencoded";
            req.ContentLength = send.Length;
            Stream sout = req.GetRequestStream();
            sout.Write(send, 0, send.Length);
            sout.Flush();
            sout.Close();
            HttpWebResponse res = (HttpWebResponse)req.GetResponse();
            StreamReader sr = new StreamReader(res.GetResponseStream());
            string responseString = sr.ReadToEnd();
            textBox1.Text = responseString;