No interviewer is going to sign an NDA for you, and asking may well earn you an immediate pass.
He does not want me to show the exact way we implemented certain parts of the user experience. I could more carefully curate which code sample I'd like to show employers.
Talk to your client and see if you can get his permission (in writing) to:
- Demo the app. Make it clear that it will be done on your device, only in the context of an interview, and that you won't distribute copies for any reason. You might also offer to delete all the copies you have once he's published the app and/or you've accepted an offer (and thus no longer need to demo it).
- Show the parts of the source that he's comfortable letting other people look at. Again, make it clear that you won't distribute copies.
These are pretty reasonable requests and you can probably work something out with your client. This way, you can show potential employers something cool that you made and the way you write code, while preserving your client's proprietary information and respecting his wishes. As long as your client is a reasonable person, you'll end up with something that you can show during your interviews.
If you get asked to show the code for the super secret, super cool bits, politely explain that it's proprietary and your contract forbids you from revealing the source. There should be no issues whatsoever with that response, but if there are, then you just dodged a bullet, because you're interviewing with a company that doesn't respect contracts, ethics, or privacy. If that happens, thank them for their time and excuse yourself.