I've never seen a dead chicken on the road, and I've lived in a very agricultural area the past eight years. If the joke was about an opossum or deer I'd understand the suicidal slant you're suggesting, but as is I'm pretty sure the "joke" is that there is no actual punch line, just a mundane response.
Add to your places to visit: San Juan Bautista, California.
I've seen several there while visiting. They have chickens all over that city wandering around, and if a cop catches you not yielding to one and flattening it, you get heavily fined. Evidently, the latter part doesn't matter.
