For something to have rights, they must be cognicent of those rights. The traditional concept of zombies does not leave much room for making such a claim. However if intelligent self aware zombies were to emerge then you might be able to make a case for it.
As it has been stated, as of existing laws -- zombies would have no rights. They are officially dead and all rights end with death. You don't have a right to life, liberty, schooling, the basic meal, or anything like that after you are dead. Have you ever seen a dead person fight for their right to do or say anything?
Even after we create new laws to deal with zeds, what are we going to do? Let's see, "Amendment X -- Barhah. No person shall kill an undead person or deprive an undead person of basic rights. These rights inclue rights stated in the first Y amendments and all amendments hereforth created." BAH! Yes, America, the land of the free.
No g** marriages, no adoption of children for g** couples, no marijuana (unless it is medical and you live in MI), prejudice, racial descrimination, and the only place where zeds can get a job. Pfffft....
Imagine: Walmart, a nice sunny day outide, and a woman is looking to buy some diapers for her newborn baby.
"Excuse me, sir, do you work here?"
THERE WOULD BE SOME ORGANIZATION THAT WOULD WANT TO FIGHT FOR ZOMBIE RIGHTS. THEY WOULD HUDDLE THEM UP IN SOME KIND OF CAMPS... HUNDREDS OF THEM, MAYBE THOUSANDS. TRYING TO FIND A CURE. THEY WOULD MAKE COMMERCIALS TITLED "SAVE THE ZOMBIES", ASK EVERY ONE TO DONATE SOME MONEY. A DOLLER HERE... 10 DOLLERS THERE.