Now, the common hydrogels are polyacrylamides; the breakdown products include acrylamide, a lethal neurotoxin. The article suggests the solar still is using polyvinyl alcohol instead of polyacrylamides; so maybe that bullet is dodged but from what we know of vinyl, this stuff won't have pleasant breakdown products either.
As Wayne points out, the water produced by the still ought to be chemically pure. But what about the sludge left behind? It's going to be full of this stuff and its breakdown products, and it's going to be a disposal problem; every time the still gets dumped, washed, cleaned, or otherwise maintained, these chemicals will be going somewhere. So I am not sanguine, even though I'm not enough of a chemist to evaluate how PVAs (polyvinyl alcohols) break down.
My first guess was they'd break down into nasty chemicals. But, if they don't, wow, what a cool thing, if it made solar stills work better.
Hope someone who is a chemist chimes in here. I think it's such a neat thing, to up the output speed... but not at the expense of gick. That's why I distill water to start with, to eliminate gick and gack.