First of all, we need to clarify the meanings of evidence. There are two main types of evidence to be considered: evidence of facts, and evidence of cause.
Absence of evidence of a fact:
Absence of evidence of the fact "that black swans exist" is not evidence that "black swans exist".
Absence of evidence of the fact "that black swans exist" is not evidence that "black swans do not exist".
Absence of evidence of the fact "that black swans do not exist" is not evidence that "black swans exist or do not exist". It is not possible to provide evidence that "X does not exist".
Absence of Evidence of Cause
Evidence of cause is much more complex. Causes are simply not simple. Every cause has a cause. Every cause of a cause has a cause. Every cause, and every consequence has a long chain of causes, as long as our imagination can create. We can often split causes into component causes, creating even longer lists of cause.
Proof of cause is subjective, not objective. Causes can be measured individually, in individual cases - or statistically - in general situations. Specific causes are subject to judgement in the individual case. Statistical causes are statistics: "lies, damn lies, and statisitcs". It is possible to demonstrate many impossible things before breakfast, using statistics.
Evidence of cause is always subjective, always subject to challenges, to appeals, to more and more complex decision processes. Lawyers thrive by creating evidence of cause and challenging evidence of cause.
Absence of evidence of cause is meaningless. It might mean there has been no search for evidence. It might mean that there has been a search for evidence of cause, or search for cause. It might refer to a trivial or cursory search for cause, or a long complex search for cause, or for evidence of cause.
But no evidence of cause has been found. That's a simple reality.
The problem we often encounter is the leap from absence of evidence to "evidence of absence".
The leap from "no evidence of cause by X has been found" to "x did not cause" is simply not logical. Innocent until proven guilty is not logical, it is moral.
Causes are not "presumed innocent until proven guilty". There is no need for a moral assumption of innocence. There is no such thing as "proof of cause" and no such thing as "proof of not cause" in the laws of science, only in courts of law.
With regards to facts, claims of an absence of evidence is nothing, proves nothing. It might be supported by evidence of a search, but does not contain evidence of a finding.
With regards to cause, claims of absence of evidence is generally an absence of imagination, or a suspension of belief.
re: Claims of Evidence
Every claim is a fact. Every fact can be evidence. All claims of evidence and all claims of absence of evidence are facts in themselves, even when the claim is false.
Any claim of evidence is "evidence" that the fact is true.
However, a claim of "there is no evidence that" is not based on evidence. It is simply a claim. In many cases, it is simply building a wall, an attempt to ignore evidence, that must be supported by denial, or renounced, if or when evidence is produced.