No rights at all. Life, liberty and the pursuit of happiness. No life, as we currently define it.
There was an entertaining TV series on this called in the flesh or something to that effect. They found a "cure" or something and all the zombies went home to their families and had to deal with he memories of what they did while still craving flesh. More than a little on the drama side.
I'm with Red. Once you're dead, you lose all your rights.
But it would be interesting to hear the Supreme Court decision: "Today, the Supreme Court ruled on the Zombie Rights Case. Justice Roberts, writing for the majority said: 'YEARRRGH!! SHE'S EATING MY BRAINS!!' Justice Sotomayor, for the minority simply stated 'BRAINS!' It will be a tough one for lower courts to interpret."
^True, BUT you must be legally declared dead by a doctor after an exam and a body identification to get a death certificate.
Tell me a zombie that is going to hang around for THAT.. after he eats his next of kin and the coroner, he's gonna head down to the mailbox to get his social security check.