The site claims that the device can provide 20 cm H₂O (which is ~1961 Pa). Apparently some hypothetical normal man breathes at a rate of 6 liters (i.e. 0.006 cubic meters) per minute at rest [1]. If you pump a volume V of fluid across a pressure difference of ΔP, you've done work V·ΔP, which is 11.76 Joules per minute or 0.1961 Watts. Keep in mind that this is the work done, not the power consumed, but let's be generous and assume a 100% efficient blower. We'll also assume that no power is harvested during exhalation.
In eight hours, that's 1.56 Watt hours.
Duracell's zinc-air chemistry claims up to 442 Watt hours / kg [2]. That's about 3.5 grams of zinc-air goo per night, ignoring packaging, blower loss, etc.
This seems entirely plausible to me.
[1] http://www.normalbreathing.com
[2] http://media.ww2.duracell.com/media/en-US/pdf/gtcl/Technical...
Edit: In the interest of precision, I should add that the analysis is accurate for an incompressible flow. Air is definitely compressible, so, if you pump air, more volume is going into the pump than is coming out, and you're heating up the air (adiabatically if your pump is any good). So, with a compressible fluid, you have to consider not on the work done moving the fluid across a pressure difference but also the energy you've stored in the fluid by pressurizing it. The pressure difference for CAP is tiny here, though, so this would be a miniscule correction to the math above.