You need to put some sort of adverb on the second part of the statement, and modify the does not. It should read something like "There is no proof that X does or does not exist, therefore I can logically conclude that X does not exist". Also it's i.e. and not e.g.
The way you have worded it, only an idiot would say false because regardless of whether it is logical to conclude that X does not exist, you CAN conclude that, unless you are brain dead and incapable of making conclusions at all.
But to answer your eg, as worded it is false. However,
Lack of evidence that something exists proves nonexsitence if the presence of the lacking evidence is a prerequisite for the thing's existence. But that is pretty much self-evident.