No., or at least no more than any other statement is unfalsifiable. For example it is easy to falsify the statement “A giant human 30, 000 km tall exists”. Such a person would be noticed with even a cursory look at Google Earth. He isn’t the ergo he has been falsified. Similarly “An elephant that can only live in my kitchen exists” can be falsified very easily, and so forth.
We could only claim those things to be unfalsifiable by altering the definition of those things. A human is a species enedemic to Earth. An elephant is a large creature made of regular matter. They can only be unfaslifiabe if we alter those definitions and propose a giant human on another planet, or an invisible elephant.
But if we want to do that then no statement is falsifiable. “A giant human 30, 000 km tall does not exist” is equally unflasifiable because even if we find a giant human of that size I can twist the defintion and say that it isn’t a giant human, rather it is a bumblebee in disguise.
The thing is that any statement can be phrased as its negative. “x exists” is exactly the same as “x does not fail to exist” or “x is not extinct”. If statements of the format “x exists” is unfalsifiable then all statements are unfalsifiable because all statements can be phrased in the format “x exists”.