Try the following:
string url = "https://api.screenleap.com/v1/screen-shares";
HttpWebRequest request = (HttpWebRequest)WebRequest.Create(url);
request.Method = "POST";
request.Headers.Add("accountId", "myAccountID");
request.Headers.Add("authtoken", "myTokenNumber");
HttpWebResponse response = (HttpWebResponse)request.GetResponse();
Stream resStream = response.GetResponseStream();
StreamReader reader = new StreamReader(resStream);
string responseFromServer = reader.ReadToEnd();
Console.WriteLine(responseFromServer);
Works for me. Hope that helps.