The modern zombie burst its way into popular culture in 1968, care of Night of the Living Dead. They didn't eat brains yet, but elsewhere at the same time, brains were blurring the real boundaries between life and death.
People whose hearts didn't pump blood and whose lungs didn't push any air around were considered dead, until 1968. With the emergence of new technologies, by hooking people up to machines, it became possible to maintain the body's basic functions and directly monitor the brain for signs of life.
The same year George Romero released his zombie movie, a committee of the Harvard Medical School released their report on "irreversible comas." They said people in comas were still alive because we could still see their brains chugging along. Even if they were never going to wake up, the choice between unplugging these patients or keeping them on life support indefinitely got a lot more complicated, and a new concept began in medical practices, brain death.
It's an issue that deals with the very definition of life. We're only alive because of the mechanical functions of our organs, but the concept of self, starting with Descartes, has become more and more centered in the brain. Twentieth century people haven't donated their brains to science like they apparently did in Victor Frankenstein's day. Our culture says that little wad of pink and gray is us, the very thing that makes us ourselves. How could we let other people experiment with that?
That's why zombies are such a twentieth century monster. Without any apparent intelligence themselves, they come at the living for the very thing they lack, the one thing no prosthetic or transplant could ever replace, our brains. Again, Romero's zombies don't eat just brains. It was Russo, co-writer of Night of the Living Dead, who thought that one up.
See, zombies aren't dead. They may not breathe or pump blood through their formerly beating hearts, but there's still trace amounts of brain activity, which is why 1968 was such a banner year. The comatose and the walking dead, they go by different names and raise different fears, but they redefined death side by side.