I need a clarification before I can answer.
You see 'evidence' is
1. (noun) 'the available body of facts or information indicating whether a belief or proposition is true or valid'.
or
2. (verb) be or show evidence of'
So what is/are the 'belief(s) or proposition(s)' you are talking about?
You said, "non-existent thing". What is that?
According to
Stanford Encyclopedia of Philosophy:
A nonexistent object is something that does not exist. Some examples often cited are: Zeus, Pegasus, Sherlock Holmes, Vulcan, the perpetual motion machine, the golden mountain, the fountain of youth, the round square, etc. Some important philosophers have thought that the very concept of a nonexistent object is contradictory (Hume) or logically ill-formed (Kant, Frege), while others (Leibniz, Meinong, the Russell of Principles of Mathematics) have embraced it wholeheartedly.
In my view the problem is that 'the non-existence of a thing' is different from 'a non-existent thing'.
As you can see, the 'non-existence' of milk in the fridge has evidence.
But... milk is not an example of a 'non-existent thing'.... is it?
I think the question is: Given a particular X, does evidence exist for the proposition that
'X is a non-existent thing'?
This is different than asking: Given a particular X, does evidence exist for the proposition that 'X has not been detected'?
Is 'a tree in the middle of a tennis court' a 'non-existent thing'?
X = 'a tree in the middle of a tennis court'
The proposition is:
'A tree in the middle of a tennis court is a non-existent thing'.
Well, I'm not sure that there is any evidence that a tree in the middle of a tennis court is a non-existent thing. I'm not sure that such evidence could be found. So, it seems to me, I would be better off looking for evidence that
'A tree in the middle of a tennis court is an existent thing'
So I would look for evidence of a tree in the middle of a tennis court.
If I found evidence that confirmed it's existence, then that would be great!
But.... can I really find evidence that a tree in the middle of a tennis court is a non-existent thing?!?
Hmm.
I could find evidence of 'the non-existence of a tree in the middle of a tennis court'. But that is not the same thing as asking for evidence that 'a tree in the middle of a tennis court is a non-existent thing'.
This is a very nuanced discussion.
For example, we also have the well-known aphorism:
'Absence of evidence is not evidence of absence'.
On the other hand:
In some circumstances it can be safely assumed that if a certain event had occurred, evidence of it could be discovered by qualified investigators. In such circumstances it is perfectly reasonable to take the absence of proof of its occurrence as positive proof of its non-occurrence.
— Copi, Introduction to Logic (1953), p. 95