Actual story is that in a socialist system like Cuba's, the internet is a state run service. The Cuban government prioritizes broadband internet access to schools and workplaces, so private internet hookups are pretty rare. Anecdotally, when I was in Cuba Cubans would crowd around public wifi hotspots for good internet access, since their ability to do so at home was lackluster at best. You can see a list of the 1000+ wifi hotspots here. That said, they definitely have internet access (so long as you have a smartphone or computer, however) and as of 2018 (so after I was there, haven't seen this on the ground myself) Cubans can get 3G and 4G internet on their phones. According to Cuban news around 7 million Cubans (a little over 60%) have internet, 4.2 million of those through mobile, and Western media backs up those numbers.

As the state has control over the internet within Cuba, it wouldn't surprise me if they cut off access to social media during times of increased dissent and foreign meddling. As the example of China shows, cultivating home grown social media vs. things like Facebook and Twitter (which can be easily astroturfed by foreign actors) is a great way to counter the traditional color revolution route pursued by Western imperialist powers. That said, Cuban internet is rickety, so it wouldn't surprise me either if a lot of the outages are just because usage spiked and then blew everything up, or a combination of both. You can even check out the usage stats here and you'll see a massive spike when the protests start. That would definitely cause an outage! Likewise, you can see form this usage data that the internet has not been "turned off" in the slightest—in fact, usage is still higher than it was before the protests began.

I know if I was an embattled communist regime next to the world's largest and most dangerous imperialist power I'd be pulling the plug on Facebook and keeping that plug pulled a la the Great Firewall indefinitely.