As a point of clarification, in standard logic, existence is not treated as a property of things, but a property of concepts. One only references a 'thing' if that thing exists. Hence, one does not say of a thing that it does not exist, but instead one says of a concept that it is uninstantiated. For example, "the king of France does not exist" would be understood to mean the concept 'king of France' has no instances, or at least no unique instances.
One might object that this is too restrictive, and indeed there are other logics, called free logics, in which the rules governing existence are more liberal. We sometimes give names to things whose existence is conjectured. In the 19th century, some astronomers conjectured the existence of a small planet inside the orbit of Mercury, and it was given the name 'Vulcan'. It was intended to explain anomalies in the orbit of Mercury. Subsequently, in the light of relativity theory, it was shown that no such conjecture was needed, so we may reasonably infer "Vulcan does not exist". This would be an example of a non-existence claim being a plausible inference on the basis that a conjectured thing has no explanatory or predictive value. It is an application of the principle of parsimony, or Ockham's Razor.
Another type of example might be where extensive searching or testing has been conducted for the existence of a thing, but it has not been found. In the 1990s some astronomers conjectured the existence of a second star in our solar system, co-orbiting with the sun. It was proposed to be a small star, perhaps a red dwarf, about one light year distant, and with an eccentric orbit that periodically carried it through the Oort cloud. It was tentatively named 'Nemesis' and its existence was intended to explain various phenomena. Astronomers have searched for this star now for many years and have not detected it. Given that we should be able to detect the existence of a red dwarf at that range, it seems reasonable to conclude that it does not exist.
Another type of example might involve the existence of things whose description is vague or highly uncertain. We might plausibly infer that 'El Dorado' does not exist, since it has never been found, but the concept is so vague, it is just unclear. Is El Dorado a man, a city, or a kingdom? How much gold would be there if we did find it? Likewise for the legendary King Arthur of the Britons. There might have been some historical leader on whom the legend is based, but I would be inclined to say that the stories about Arthur are too fantastical to be true, so it would be reasonable to infer that Arthur did not exist.
The examples given above are abductive inferences. We conclude that there is no good reason to suppose the existence of something, but we cannot prove it. There are examples of deductive inferences of non-existence, which take the form of showing that if a thing exists then a contradiction would follow, hence by reductio no such thing exists. Mathematics contains proofs of this kind, e.g. there is no such thing as the largest prime number.
As to your comment about the null hypothesis, I would recommend not putting much weight on the concept. The idea that we should accept some position by default in the absence of evidence is at the least controversial, and at worse downright foolish. Null Hypothesis Signficance Testing, although common, is hugely overrated and overused. You might care to consult the book: The Cult of Statistical Significance by Ziliak and McCloskey. You should definitely read a recent article by Amrhein, Greenland and McShane, published in Nature, in which the authors, together with more than 800 other signatories, call for a restriction on the use of significance testing and confidence intervals.