If the Mayans had at least gotten the shape of the planet right (or actually thought the world, and not just their calendar, would come to an end), maybe we might have reason to be concerned, but still, a critical thinker should be thinking of lots of questions to scrutinize this pronouncement, first among which ought to be the most basic of all: how could the Mayans (or anyone else, really) possibly be justified in making such a prediction about the future in the first place?
As you may expect, this is one of those situations in which some vague and broad pronouncement by some mysterious source only gets its details filled in by those who subsequently believe it. In other words, massive cognitive biases. Being a believer is easy, but the line between belief and gullibility is thin. If you want to think critically, the way to do it is not to simply accept confirmatory evidence, however vague and dubious, but to look for some way to falsify the pronouncement, and then see what pans out.
And Neil deGrasse Tyson (maybe drunk?) has a few more lessons to dish out: