Page 4 of the 5e Monster Manual, under the title 'What Is a Monster', it says:-
"A monster is defined as any creature (emphasis added) that can be interacted with and potentially fought and killed. Even something as harmless as a frog or as benevolent as a unicorn is a monster by this definition. The term also applies to humans, elves, dwarves, and other civilised folk who might be friends or rivals to the player characters."
Therefore, all monsters are creatures, but not all creatures are monsters (in the same way that all poodles are dogs but not all dogs are poodles).
So what is the difference between creatures that are monsters, and creatures that are not monsters?
Well, since 'creatures that are monsters' are 'creatures that can be interacted with and potentially fought and killed', then 'creatures that are not monsters' are 'creatures that cannot be interacted with or fought and potentially killed'.
Since 'creature' is not directly defined, and since 5e proudly uses 'natural language', then without a game definition we must assume that 'creature' has its natural language meaning:-
Dictionary.com: "noun
1 an animal, especially a nonhuman
2 anything created, whether animate or inanmate
3 person, human being
4 an animate being"
I'm confident that a chicken can be interacted with, and potentially fought and killed. I'm confident that a chicken satisfies one or more of the dictionary definitions of the word 'creature'.
The hex spell, like most spells, targets a 'creature'. The player, knowing that by the rules in the MM, by the dictionary definition, and by natural language, knows that a chicken is a creature.
The DM says otherwise. Not because he believes it, or that he can point to a rule which shows otherwise, or could point to a dictionary which shows that chickens cannot be described as a creature, but because the DM doesn't like how the player is using a spell.
Is the player being unreasonable about what 'creature' means, or is it the DM who's being unreasonable.
Note that if the DM insists that chickens are not creatures, then he needs to define what definition of the word 'creature' he is using for 5e. Then, that definition remains the same no matter the intentions or actions of any player or PC.
Oh, chickens aren't creatures? Okay, chickens are immune to fireball. "Each creature in a 20-foot radius must make a Dexterity saving throw. A target takes 8d6 fire damage on a failed save, or half as much damage on a successful one."
Shapechange: "You assume the form of a different creature for the duration". If chickens aren't creatures, then you can;t shapechange into one.
Power Word: Kill: "You utter a word of power that compel one creature you can see within range to die instantly". Chickens are immune to PWK.
Why don't chickens rule the multiverse?
Anyway, I hex a puppy. Why? I'm conducting an in game experiment about which creatures are creatures and which aren't.
If the DM says that chickens aren't creatures therefore cannot be hexed, then EITHER he is screwing me over, actually lying to me, OR he has such an esoteric definition of 'creature' that it would change his gameworld into a total farce, with PCs charging into combat in their chicken-suit armour, with fireballs and PWKs bouncing off.
I though the motive for 'ruling' that chickens are not 'creatures' was to avoid absurdity?