I'm not sure why you can't handle HTTPS. I wrote a helper class for my own applications and I am able to GET/POST to HTTPS without a problem. I will post the code here and perhaps you can see if there are differences between my code and yours.

import java.io.IOException;
import java.io.InputStream;
import java.io.UnsupportedEncodingException;
import java.net.HttpURLConnection;
import java.net.URL;
import java.net.URLConnection;

import org.apache.http.HttpResponse;
import org.apache.http.client.methods.HttpGet;
import org.apache.http.client.methods.HttpPost;
import org.apache.http.client.params.ClientPNames;
import org.apache.http.client.params.CookiePolicy;
import org.apache.http.entity.StringEntity;
import org.apache.http.impl.client.DefaultHttpClient;
import org.apache.http.params.BasicHttpParams;
import org.apache.http.params.HttpConnectionParams;
import org.apache.http.params.HttpParams;
import org.apache.http.protocol.BasicHttpContext;
import org.apache.http.protocol.HttpContext;
import org.apache.http.util.EntityUtils;
import org.json.JSONObject;

import android.util.Log;

public class HttpRequest{

    DefaultHttpClient httpClient;
    HttpContext localContext;
    private String ret;

    HttpResponse response = null;
    HttpPost httpPost = null;
    HttpGet httpGet = null;

    public HttpRequest(){
        HttpParams myParams = new BasicHttpParams();

        HttpConnectionParams.setConnectionTimeout(myParams, 10000);
        HttpConnectionParams.setSoTimeout(myParams, 10000);
        httpClient = new DefaultHttpClient(myParams);       
        localContext = new BasicHttpContext();    
    }

    public void clearCookies() {
        httpClient.getCookieStore().clear();
    }

    public void abort() {
        try {
            if (httpClient != null) {
                System.out.println("Abort.");
                httpPost.abort();
            }
        } catch (Exception e) {
            System.out.println("Your App Name Here" + e);
        }
    }

    public String sendPost(String url, String data) {
        return sendPost(url, data, null);
    }

    public String sendJSONPost(String url, JSONObject data) {
        return sendPost(url, data.toString(), "application/json");
    }

    public String sendPost(String url, String data, String contentType) {
        ret = null;

        httpClient.getParams().setParameter(ClientPNames.COOKIE_POLICY, CookiePolicy.RFC_2109);

        httpPost = new HttpPost(url);
        response = null;

        StringEntity tmp = null;        

        Log.d("Your App Name Here", "Setting httpPost headers");

        httpPost.setHeader("User-Agent", "SET YOUR USER AGENT STRING HERE");
        httpPost.setHeader("Accept", "text/html,application/xml,application/xhtml+xml,text/html;q=0.9,text/plain;q=0.8,image/png,*/*;q=0.5");

        if (contentType != null) {
            httpPost.setHeader("Content-Type", contentType);
        } else {
            httpPost.setHeader("Content-Type", "application/x-www-form-urlencoded");
        }

        try {
            tmp = new StringEntity(data,"UTF-8");
        } catch (UnsupportedEncodingException e) {
            Log.e("Your App Name Here", "HttpUtils : UnsupportedEncodingException : "+e);
        }

        httpPost.setEntity(tmp);

        Log.d("Your App Name Here", url + "?" + data);

        try {
            response = httpClient.execute(httpPost,localContext);

            if (response != null) {
                ret = EntityUtils.toString(response.getEntity());
            }
        } catch (Exception e) {
            Log.e("Your App Name Here", "HttpUtils: " + e);
        }

        Log.d("Your App Name Here", "Returning value:" + ret);

        return ret;
    }

    public String sendGet(String url) {
        httpGet = new HttpGet(url);  

        try {
            response = httpClient.execute(httpGet);  
        } catch (Exception e) {
            Log.e("Your App Name Here", e.getMessage());
        }

        //int status = response.getStatusLine().getStatusCode();  

        // we assume that the response body contains the error message  
        try {
            ret = EntityUtils.toString(response.getEntity());  
        } catch (IOException e) {
            Log.e("Your App Name Here", e.getMessage());
        }

        return ret;
    }

    public InputStream getHttpStream(String urlString) throws IOException {
        InputStream in = null;
        int response = -1;

        URL url = new URL(urlString); 
        URLConnection conn = url.openConnection();

        if (!(conn instanceof HttpURLConnection))                     
            throw new IOException("Not an HTTP connection");

        try{
            HttpURLConnection httpConn = (HttpURLConnection) conn;
            httpConn.setAllowUserInteraction(false);
            httpConn.setInstanceFollowRedirects(true);
            httpConn.setRequestMethod("GET");
            httpConn.connect(); 

            response = httpConn.getResponseCode();                 

            if (response == HttpURLConnection.HTTP_OK) {
                in = httpConn.getInputStream();                                 
            }                     
        } catch (Exception e) {
            throw new IOException("Error connecting");            
        } // end try-catch

        return in;     
    }
}

As some of the methods are deprecated, should'nt it be like this?

  private DefaultHttpClient createHttpClient() {
    HttpParams params = new BasicHttpParams();

    HttpConnectionParams.setConnectionTimeout(params, 10000);
    HttpConnectionParams.setSoTimeout(params, 10000);
    HttpProtocolParams.setVersion(params, HttpVersion.HTTP_1_1);
    HttpProtocolParams.setContentCharset(params, HTTP.DEFAULT_CONTENT_CHARSET);
    HttpProtocolParams.setUseExpectContinue(params, true);

    SchemeRegistry schReg = new SchemeRegistry();
    schReg.register(new Scheme("http", 80, PlainSocketFactory.getSocketFactory()));
    schReg.register(new Scheme("https", 443, SSLSocketFactory.getSocketFactory()));
    ClientConnectionManager conMgr = new PoolingClientConnectionManager(schReg);

    return new DefaultHttpClient(conMgr, params);
  }

Should I change anything else, like HttpVersion?