Q:

Does the Protection from Evil and Good spell give an undead disadvantage to attacking itself? 

If a zombie attempts suicide by attacking itself while under the effect of a protection from evil and good spell, would it have disadvantage on the roll to kill itself? 

A:

Rules as Written (RAW), Yes
Protection From Evil and Good states:

Until the spell ends, one willing creature you touch is protected
  against certain types of creatures: aberrations, celestials,
  elementals, fey, fiends, and undead. 
The protection grants several benefits.  Creatures of those types have disadvantage on attack rolls against the target. ( PHB, p. 270, bold added)

There is nothing in this description that indicates that the target of the spell must be different from the attacker.  So yes, this spell would grant disadvantage on such an attack. 
Note that this would rarely come up, especially since the target must be willing to have the spell cast upon themselves in the first place.  Also note that if the target of the spell gave itself advantage somehow (such as by going prone, which would give it both advantage and disadvantage), it would just roll normally.  And a DM could hypothetically decide that an honest attempt to injure oneself is automatically successful.  But RAW yes, technically this spell could make it more difficult for a creature to strike itself. 