In metaphysics, "Humeans" are people who believe that truths about laws of nature, counterfactuals, dispositions and the like (truths about what must or would be the case) are in some sense reducible to non-modal truths (about what is the case).
One way to be a Humean is to deny that there are any laws of natures, non-trivial counterfactuals, etc.: if there are no modal truths, then trivially all modal truths are reducible to non-modal truths. On this account, there are no "necessary connections between distinct existences": eating arsenic might in fact be followed by death, but it could just as well be followed by hiccups or anything else.
I don't know anybody who holds that view. At any rate, a lot of Humeans believe in laws of nature, counterfactuals, and all that stuff. But they also believe that these modal facts are made true by non-modal facts, perhaps about regularities in the distribution of categorical properties. On this view, there are necessary connections between distinct existences. It may well be (nomologically) necessary that consuming large amounts of arsenic leads to death.
Unfortunately, some people keep portraying Humeanism as the denial of necessary connections. Please don't do that. It's like saying that physicalism is the view that there are no mental properties. Eliminativisism about the mental is a particular kind of physicalism (a rather unpopular kind). Similarly, eliminativism about necessary connections is just one kind of Humeanism -- a kind that hardly anyone believes in.
The association of Humeanism with the rejection of necessary connections is especially popular among anti-Humeans. For example, Galen Strawson suggests in his book "The secret connexion" that according to the regularity theory of causation,
the regularity of the sequence of events in the world as a whole is just like the regularity of the sequence of natural numbers 0,1,2,3,4,5,..., in the extraordinary but obviously logically possible case in which the sequence is being generated by a true random number generator. (p.24)
Analogies like this are a standard tool in anti-Humean rhetoric. But they have little force against Humeans who believe in laws of nature. If there is a law of nature according to which state 1 is followed by 2, 2 by 3, etc., then in what sense is the resulting sequence like the output of a random number generator? If the laws are deterministic, then state 1 must -- with nomological necessity -- be followed by state 2. No such necessary connection exists between the numbers generated by a random number generator. To accept a nomologically necessary connection between state 1 and state 2, all you need is to accept the relevant law. It doesn't matter if you also think the law is reducible to regularities.
Now some non-Humeans believe that certain successions of events in our world are not merely nomically necessary, but absolutely, metaphysically necessary. Humeans typically don't believe that: it is not metaphysically necessary, on their view, that consuming arsenic leads to death. So could we characterise Humeanism as the denial of metaphysically necessary connections between distinct existences?
This proposal at least has the advantage that some people actually hold the relevant view. However, many people who are commonly classified as non-Humeans also hold that view. To believe that truths concerning the laws of nature are irreducible to non-modal truths is not the same as to believe that the laws of nature hold with metaphysical necessity.
Another difficulty with this proposal is to clarify "distinct existences". Humeans like Lewis don't deny that e.g. people may have there origins essentially, and so that there are metaphysically necessary connections between distinct people. Similarly, Lewis would have happily accepted metaphysically necessary connections between distinct properties: perhaps the property of consuming arsenic in a world with the actual laws of nature is connected by metaphysical necessity to the property of dying, and I suppose these two properties are distinct (whatever that exactly means -- it better not mean that the two properties are not connected by metaphysical necessity: this would make the supposedly Humean doctrine analytic).
One might try to restrict the "distinct existences" to intrinsic properties, or express the denial of necessary connections in terms of intrinsic duplication. Unfortunately, "intrinsic" is used in many different ways, and we would have to make sure we're not talking past each other. For example, it seems to me that Lewis means something quite different by "intrinsic" than contemporary necessitarians about laws. For Lewis, an intrinisc property is essentially a property P such that a thing X having P entails nothing at all about the rest of the universe. Necessitarians generally do not think that properties like mass and charge are intrinsic in this sense, although they often call such properties "intrinsic". (I am not sure what they mean by that.)
This is not to say that there is no interesting distinction between various forms of necessitarianism and their alternatives. For example, necessitarians tend to regard causation as Lewis-intrinsic, and mass and charge as Lewis-extrinsic, which is opposite to Lewis's own account. But this somewhat delicate debate about the intrinsicality of various properties is only very indirectly related to the question of whether nomic truths are reducible to categorical truths.