user3791884,
Any luck with your C# PUT? Here is C# code that works: 
    using System.Net.Http;
private async void changeAway()
{
    using (HttpClient http = new HttpClient()) 
    {
        string url = "https://developer-api.nest.com/structures/" + structure.structure_id + "/?auth=" + AccessToken;
        StringContent content = new StringContent("{\"away\":\"home\"}"); // derived from HttpContent
        HttpResponseMessage rep = await http.PutAsync(new Uri(url), content);
        if (null != rep)
        {
            Debug.WriteLine("http.PutAsync2=" + rep.ToString());
        }
    }
}
Debug.WriteLine writes this to the Output window:
"http.PutAsync2=StatusCode: 200, ReasonPhrase: 'OK', Version: 1.1, Content: System.Net.Http.StreamContent, Headers:
{
  Access-Control-Allow-Origin: *
  Cache-Control: no-cache, max-age=0, private
  Content-Length: 15
  Content-Type: application/json; charset=UTF-8
}"
These two methods return a valid structure of my data.
1/ command line
curl -v -k -L -X GET "https://developer-api.nest.com/structures/Za6hCZpmt4g6mBTaaA96yuY87lzLtsucYjbxW_b_thAuJJ7oUOelKA/?auth=c.om2...AeiE"
2/ C#
private bool getStructureInfo()
{
    bool success = false;
    try
    {
        // Create a new HttpWebRequest Object to the devices URL.
        HttpWebRequest myHttpWebRequest=(HttpWebRequest)WebRequest.Create("https://developer-api.nest.com/structures/?auth=" + AccessToken);
        // Define the request access method.
        myHttpWebRequest.Method = "GET";
        myHttpWebRequest.MaximumAutomaticRedirections=3;
        myHttpWebRequest.AllowAutoRedirect=true;
        myHttpWebRequest.Credentials = CredentialCache.DefaultCredentials;
        using(HttpWebResponse myHttpWebResponse = (HttpWebResponse)myHttpWebRequest.GetResponse())
        {
            if (null != myHttpWebResponse)
            {
                // Store the response.
                Stream sData = myHttpWebResponse.GetResponseStream();
                // Pipes the stream to a higher level stream reader with the required encoding format. 
                StreamReader readStream = new StreamReader (sData, Encoding.UTF8);
                Debug.WriteLine("Response Structure stream received.");
                string data = readStream.ReadToEnd();
                Debug.WriteLine(data);
                readStream.Close();
                success = deserializeStructure(data);
            }
        }
    }
    catch (Exception ex)
    {
        Debug.WriteLine("getStructure Exception=" + ex.ToString());
    }
    return success;
}