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?
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.
Technically, yes, but would the zombie do that?
I think the rules for Protection Against Evil and Good are fairly clear in how they work and the answer is yes. The zombie, which is undead, is attacking itself and therefore has disadvantage.
But it's worth asking: why would a zombie attack itself? They're supposed to be mindless animated corpses with very simple motivations: kill the living.
For a zombie to think to kill itself means that it has to be capable of some pretty high level thought processes (sense of self, goals, cost/benefit analysis, etc) and these aren't really in keeping with the lore of zombies. Obviously, you can do whatever you want with your games, but I think ZAI (Zombies as Intended), a zombie wouldn't attack itself. A zombie has an Intelligence score of 3.
A zombie can follow simple orders and distinguish Friends from foes, but its ability to reason is limited to shambling in whatever direction it is pointed, pummeling any enemy in its path. A zombie armed with a weapon uses it, but the zombie won’t retrieve a dropped weapon or other tool until told to do so. (MM, p. 315, Zombie)