First I'll say that without preregistration of the methodology, there's a lot that is immediately suspicious.
> The researchers built an index based on 15 relevant questions in the NCHA, in which students were asked about their mental health in the past year
Why these 15? What was the "relevance" criteria?
To their credit, they don't just look at a summary metric of "mental health" which would be kind of absurd since the relative weighting is also arbitrary (although that appears to be the main conclusion). The article here notes several axes on which significant differences were found. Why these axes? What about other "mental health" metrics? Did they get better or stay neutral or just have no detectable effect?
Without preregistration it's almost impossible to determine exactly how cherry-picked these differences were, as with a large enough set of potential questions to choose from, you're going to find statistically significant trends on some of them by random chance.
The core methodology is to track the spread of Facebook to different colleges and compare mental health between schools that had Facebook and schools that did not yet have Facebook. This is surprisingly not terrible, but without insight into how the study controlled for the time axis and potential confounding variables about the non-random selection of schools for the rollout, it's difficult to say more.