You can't really do this. Your application can be disassembled and whatever secret is in the binary can be replicated in a malicious application.
Another attack you should be aware of is people settings the hosts file to a location they control and then installing a root certificate that allows them to provide a signature for that domain. Your application would do the post with the secret, and they'd just be able to read out the secret. They could extract the password from any complicated encryption system within the binary in this way.
Most of the ideas in this thread are vulnerable to this attack.
That said, the likelihood of somebody caring enough to disassemble your application is probably fairly remote.
I'd just keep it simple. Have a password that's hardcoded in to your application. To prevent someone just looking at the resources and trying every string, make it the XOR of two strings or the result of an AES decrypt of a particular fixed string.
Obviously, you should do the request over SSL otherwise an attacker can just sniff the traffic.
Yes, a determined attacker will circumvent the scheme but like any DRM scheme, that's always been the case. The trick is to make it too much effort to be worth it.
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