And here was an answer that completely blew my mind:
Basically, Reddit user l3mon_aid is saying that the joke is actually about the chicken killing himself.
OR IS IT? Other commenters argued that the joke is not anywhere near as morbid as all that. That it's just a dumb non-joke: The chicken crossed the road to get to the other side. Of the road. The same reason anyone would cross a road.
But now I can't stop thinking about the dark and depressing chicken-crossing-the-road joke. What if that's actually what the joke is really about?