Not sure what they're claiming. watched their promo vid and I'm just more confused.
They spend a lot of time talking about how their phone is all about privacy, and I would agree that most android devices are not user or privacy focused (looking at you uninstallable Sprint Nascar app).
That said, Apple's iOS is already a fairly user and privacy focused device. FaceTime/Messages data is encrypted over Apple's servers. So is iCloud. Their source code is proprietary and a trade secret unlike Android so I'd wager Blackphone having their core based around android could be susceptible to some Android related vulnerabilities. They encrypt data but again, iOS already does this.
Blackberry may be fairly secure too, but I only know about iOS b/c I've had to develop apps for the platform in the past.
TLDR no such thing as legitimately secure.