Just to add a few comments to those from Syed:
Indeed your phone has a satellite receiver, but this is a specific receiver intended for receiving the GPS satellites' positioning information. You would not be able to repurpose this receiver in code to receive Outernet signals. As Syed has said, the current Outernet implementation uses Ku-Band geostationary satellites, and even were Outernet in the future to launch its own constellation of small satellites (as it plans to do), these would not transmit signals that were on the same frequency as, nor compatible with, your phone's GPS receiver.
The SPOT Connect app does not turn your phone into a satellite-capable device. It couldn't do, as your phone has no built-in satellite communications transmitter/receiver. What this app does is to use the phone's Bluetooth radio to connect to an external SPOT Connect device. This external box does contain the necessary satellite communications transmitter/receiver, which - as Syed mentioned - communicates with the Globalstar satellite constellation. The app is nothing more than an interface to the external SPOT Connect box.
As you'll see from my comments above, there is no way, therefore, that you'll be able to reverse engineer the SPOT Connect app, or write new code, that will allow your phone to directly connect to the Outernet service in the way that you propose.
However... what I do expect that Outernet will do in the future is to offer a service rather similar to the way that SPOT Connect is implemented. Once Outernet manages to move on to its next phase, and build and launch its own constellation of small, low orbiting satellites, then I am sure that there will be an "Outernet Connect" external box, that you can access via your phone so as to receive the Outernet service that way.
Unfortunately, this is not very likely. Outernet is planning only to implement a one-way, receive-only, non-real-time data service, so you'll never (well, never is a long time - I suppose it could happen one day, but not soon!) be able to use Outernet to make voice calls, browse the web, transmit data or anything that requires a two-way, transmit/receive, real-time data channel. The principal reason is cost. To implement the two-way, real-time service would cost billions of dollars. Outernet is trying to bring free access to news and information, to those in the World that currently have nothing, for a fraction of this cost.
Just to end by saying that I fully agree with Syed. New ideas and innovative thinking is great. My comments are meant to be helpful and constructive and not to squash new thinking!