As far as I can tell, the Synth gains the armor and natural armor of the eidolon. By that, if the eidolon has armor bonus of +0, and +X natural, you get to add +0 to your armor and +X to natural armor, in addition to any natural and normal armor you have.
That is at least how I read it, and if you can use shields, you can use armor
That is at least how I read it, and if you can use shields, you can use armor