Just taking the article’s title at face value, what does it mean to have a blackout with microgrids?

Microgrids are, by definition, impervious to blackouts.

If a microgrid loses power due to a fault, neighbouring grids are unaffected. If all the neighbouring grids lose power due to a common fault, but your local grid is unaffected due to design or implementation choices, you’re golden.

Microgrids aren’t a solution to blackouts, and blackouts are not an issue microgrids have.

Looking at the article, the first paragraph claims:

When power went out across all of Puerto Rico on 16 April [well, clearly it didn’t as the very next sentence goes on to claim] a lot of the lights in the town of Adjuntas stayed on.

It can’t be both. It’s not a blackout and the lights stayed on.

Grid segmentation and multiple generation sources can do this too, and is a common feature of existing, traditional, power grids. The city I live in has these features with feed-ins from multiple hydro electric plants and wind factories, and a HVDC link to the next state over which has a full spectrum of generation sources bar nuclear.

As a result, the electricity here is very stable, and I don’t recall the last time we had even a brown out that affected the entire greater metro are and satellite towns.

Microgrids with interconnects are grids.

We can build grids that work, thereby leaving the general population free to pursue other, more important, economic endeavours.

Failing that, build microgrids.