Prime places besides the smattering of isolated counties and the mentioning of Appalachia in the article: Mountain and shoreside Maine: Probably because there are lots of cabins in those parts. Southwest and East Alabama: Poor rural African Americans. The counties with major cities are the palest. Southern Texas: Poor illegal immigrants. The legal immigrants can likely afford plumbing. The Southwest, a line down the middle of Wisconsin, all of Alaska, NE Washington, south central Oregon, and bits of Montana: Probably a lot of poor Native Americans. Big Island Hawaii: I got nothing. Possibly a mixture of cabins and poor Hawaiians (they have an issue with poverty there, I think). A lot of these are also in places where it's very hard to establish plumbing.
Big Island Hawaii is largely National Park land; I imagine there's not enough room to legally build up the infrastructure that makes indoor plumbing viable. Have to hit a tipping point with utilities before they make sense. I don't know what's going on in Alaska, though. Juneau and Anchorage should skew that data enough that there are a couple of white census areas.
True; there's also the fact that the entire island is a county, and I don't know what they're counting as a 'housing unit'. Is it a dormitory used by researchers? Is it a permanent residence? Big Island is the entire above-water portion of five volcanoes, three of which are active, and I'd estimate that there's some housing used to house volcanologists and naturalists for research purposes, but I don't think even then that that would make a huge dent in the number of buildings without indoor plumbing. Also, establishing plumbing on top of a volcano without care would probably be a problem. You don't want lava burning your butt while you're having a poop. EDIT: There's also Kauai and Niihau! Niihau is a privately-owned island where a bunch of native Hawaiians are essentially doing it like their ancestors did, and Kauai has frighteningly high levels of economic inequality (the average house costs $528000 but the median household income is $62531). Looking at the map of those areas Anchorage and Juneau are the only areas that aren't colored dark blue. I imagine there's not enough room to legally build up the infrastructure that makes indoor plumbing viable
Juneau and Anchorage should skew that data enough that there are a couple of white census areas.
Good old lying with statistics. That's the best kind of lying.