A search of some of the media sources listed showed no results for either the company or the disposable phone.
Carbon is conductive so the idea of printing circuits on paper is valid.
I don't know what the power constraints would be. This type of circuit would be slow because of its size if nothing else. And flexing the paper would probably cause fractures in the circuit. But it should be possible to design a slow circuit that can be printed on a laser printer. I doubt it would be of much use in a phone though, and there is probably proprietary encoding needed to get onto the existing phone systems.