What do zombies eat? The short answer is zombies eat people. They're looking for living flesh to chew on, to swallow, to digest. In reality in a real zombie outbreak, zombies may actually not eat anything, at all; they may just bite and chew. Turns out that swallowing is actually a complex neuromuscular activity that they may not be able to accomplish.
Also, the notion of a zombie actually digesting the food it eats doesn't really hold water. Have you ever seen a zombie on the crapper? Whether they swallow and digest what they bite, or if they just chew and spit it out, it doesn't really matter, the end result is the same. What zombies attack spreads the infection, so being bitten by a zombie turns you into a zombie.
From the moment the zombies were first created in 'Night of the Living Dead', in 1968, they hungered after one thing, and often one thing only, living human flesh. That's what they want. That's what they hunt.