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?