Alright, so, I’m a pretty new dm and I have this player who really likes using Magic Hand as one of their cantrips. Basically what happens whenever I give his party an NPC that they need to fight, he says something along the lines of “I cast Magic Hand and squeeze his heart/brain.” Whenever I tell him he can’t use magic hand on the NPCs heart he gets very upset and argues with me. 
Granted a heart (Still alive, in someone’s chest) probably isn’t considered an object, am I in the wrong here for telling him he can’t send people into cardiac arrest with Magic Hand? Or is a heart considered an object even though it is alive?

The mage hand can't attack nor squeeze the heart of a creature.

Spells in 5e do what they say they do.  NOTHING else unless you as the DM say so.

"You can use your action to control the hand. You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial. You can move the hand up to 30 feet each time you use it."

That's it.  That's all the hand can do.  It's worth noting that a heart is not an object while it's inside a creature.  It's part of the creature, which is then an attack, and it doesn't say it can attack. 

One, Mage Hand, as previously mentioned, cannot be used to attack.  At all.

Second, "inside a creature" is not a valid location to manifest an effect.

Third, you can't attack specific body parts on an enemy.

The hand can't attack.

That's a great way to say "No." If it can't even punch someone for damage, then killing them by squeezing their heart is even more invalid.

This is a case of trying to make a cantrip vastly more powerful than its level. Similar to "I use create or destroy water to drown the dragon by filling his lungs with water." It's a 'clever' usage that's 100% not RAW.

Setting aside the (completely valid) objections based on the rules, I wouldn't allow it because it's bad gaming. It's a free resource, giving it the ability to kill, which makes it potentially as deadly as endgame spells. That's really unfair to give it that kind of strength. Even if the rules allowed it (they don't), I'd rule against it anyway.

DM: "Do you really want me to allow mage hand to squeeze hearts and brains for instant kills?"

Player: "Yes."

DM: "Okie dokie. Suddenly you start speaking gibberish and fall to the floor, incapacitated, convulsing. Blood is coming from your ears, eyes and nose."

Player: "Do I get a save or anything?" 

DM: "Nope. And... Now you're dead. Your brain is completely destroyed, nothing short of True Resurrection can bring you back."

Player: "But how?!"

DM: "Oh there's a 7th level Arcane Trickster rogue somewhere. They're invisible as is their mage hand."

Other Player: "I will..."

DM: "You can't do anything as the same thing starts happening to you..."

DM: "Wow, thanks for allowing this. Easiest legit TPK ever."

Players: "New char-"

DM: "Same thing will just happen to them. Invisibility and mage hands are easy to come by. You have rendered it basically pointless to play D&D. Have a great day."

I mentioned this spell in another thread but you might want to homebrew Clutch of Orcus from 3e, that should give the player all the heart grasping action they need, A rough version of it would be:

Name: Clutch of Orcus. School: Necromancy. Level 3, Components: V, S: Casting Time: 1 action. Range: 120ft. Target: One Humanoid Creature. Duration: Concentration (max 1min). Saving Throw: Wisdom.
The caster creates a magical force that grips the subject's heart (or similar vital organ) and begins crushing it. The victim is paralyzed (as if having a heart attack) and takes 1d3 points of Necrotic damage per round. In addition, a conscious victim gains a new saving throw at the end of each of its turns, ending the spell on a succesful saving throw. If the victim dies as a result of this spell, his chest ruptures and bursts, and his smoking heart appears in the caster's hand.