My question is two-fold.
First, I wonder whether a Homunculus Servant can use magic items, especially if they require attunement.
It explicitly counts as a creature of the construct type, and has an Intelligence score of 10. It cannot speak, so of course it couldn't use magic items that require a command word, but the rules for attunement state (DMG, p. 138):
Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it […]
It sounds like you could just have the homunculus take a short rest and command it to attune to an item.
I would like an answer to that question alone, along with the other question in the title. Due to the unique phrasing of the hit points formula in the Homunculus Servant statblock (TCE, p. 22), it is unclear whether increasing its Constitution from an Amulet of Health (or otherwise) would in fact actually increase its hit points:
Hit Points 1 + your Intelligence modifier + your artificer level (the homunculus has a number of Hit Dice [d4s] equal to
your artificer level)
Which seems to not include the results of the d4 hit dice themselves, but rather only include the +1 Constitution modifier from the statblock. My suspicion is that the "+ artificer level" component of the formula comes from Constitution, but maybe that's not even relevant. Maybe it's enough to say that hit points increase when Constitution does. Per the rules on hit points for monsters (MM, p. 7):
A monster's Constitution modifier also affects the number of hit points it has. Its Constitution modifier is multiplied by the number of Hit Dice it possesses, and the result is added to its hit points.
Does the statblock count as a case of "specific beats general"? Or is the Monster Manual's rule on Constitution general enough to apply in this case, since they would technically not be mutually exclusive?
Best Answer
Can a homunculus servant use magic items?
Yes.
As you note, a homunculus servant is a creature, as noted in its description, where it is referred to several times as a creature.
In general, a creature can use a magic item, subject to the limitations of that magic item, and the limitations of the creature's physique. The basic rules say, "In most cases, a magic item that's meant to be worn can fit a creature regardless of size or build", which implies that creatures of all types can use magic items.
Can a homunculus servant attune to a magic item?
Yes, as long as it can actually use the item, and it meets the item's class or other restrictions to attunement.
See this question: Can a familiar attune and wear or use a magic item?
Note, in particular, that question's discussion of D&D Adventurer's League restrictions, that the attuned items of a creature under your control count toward your attunement slots.
Personal opinion, any sensible DM is going to implement this restriction on attunement; otherwise, anyone with a companion effectively gets way more attunement slots than intended.
The DMG says (emphasis added):
Can the homunculus servant attune to an Amulet of Health?
Yes, as described above.
Does the amulet increase the homunculus' hit points?
No, although it otherwise benefits from the increased constitution, for instance, on saving throws.
The homunculus' hit point total is specifically tied, not to the constitution of the creature, but to the creator's intelligence and level.
Compare to the steel defender, which is similar.
There are other creatures with some similarities to the homunculus with special hit point rules, where hit points are not determined by the creature's constitution; for instance, the elemental spirit from the summon elemental spell.
Would it wreck the game, if the homunculus' HP were instead calculated by HD and constitution? Probably not, and it might be cool.
Talk to your DM
Of course, the DM can decide in any direction, and overrule any of this, and probably has specific opinions on all of these issues. In the end, what is balanced and fair, and makes the most fun for everyone is more important than any of this stuff.
Summary
Sticking a magic item on the artificer's little servant is cool, and allows a lot of fun possibilities.
Allowing it to attune, but to count against the creator's slots is flexible, but also bounded, keeping the character from becoming overpowered.
Keeping the HP calculation as stated in the stat block is in keeping with the rules as written and intended; but allowing the opposite would not be game-breaking, as long as it costs the creator an attunement slot.