If I understand your question correctly, you know how to do the POST in Android, but you want a "secure" way to do it. I think what you are looking for is a basic method of authentication.
The simplest way is to have a user log in with username/password (which the username can be stored in SharedPreferences or some other manner). As one user has pointed out, there ARE concerns regarding storing of passwords in SharedPreferences, though multitudes of other answers here and apps use this method of storage for it - just using the password ONCE would work until the token expires. The server app would provide this token after the first successful authentication. You'd have to track to tokens in your database and assign an expiry date to them for better security. All of this should ONLY be performed over SSL, so the POST payloads are encrypted.
Decompiling an app and figuring out an API endpoint from a request are two totally different things. For your situation, you don't have to really worry about obscuring your endpoints, only securing the POST (JSON) payload and providing a simple method of authorization. Your endpoints are going to be probed regardless, so it is best to simply put authentication mechanisms in place so only those users authorized will see the data.
Creating a Sync Adapter would help in managing all of the authentication work, and the user's USERNAME (and the token) could be saved relatively safely in Shared Preferences. You would have to, however, set up this username/password/token authentication system on your server. Here is an example process flow:
User [username/password] -> Server [authentication] -> Server [token] -> User [store token]
Then, after that, the app will authenticate against the endpoint with their token until it expires (the server will notify the app when it expires):
User [token] -> Server [validates token] -> User [data] -> Server [data]
The token could be something as simple as a random GUID.
This will ensure the user is authorized to receive the data and should ONLY be performed through a POST request and ONLY over SSL to ensure the JSON payload is as secure as possible from prying eyes. You can go into a whole bunch of other areas here, such as rate-limiting requests to prevent people bombarding your APIs with brute-force attacks or something else, but that would all be done on the server side.
Granted, this is NOT the most secure way to do this, as there are better protocols like OAuth, but they are also much more complex to implement for a simple client/server app. I would highly recommend you investigate them, however.
Sources:
Creating a Sync Adapter: http://developer.android.com/training/sync-adapters/creating-sync-adapter.html
How to do an HTTP POST in Android: How to do a HTTP Post in Android?
PHP GUID: http://php.net/manual/en/function.com-create-guid.php