First: These symbols are not standardized. You will find all kinds of "end of ..." symbols, say, black squares, white squares, black triangles, or any symbol that's contained in a dingbat font. Some authors distinguish between the symbols for "end of proof", "end of example", and "end of definition", some authors use the same symbol everywhere, and some authors mark only the end of proofs, but not the end of examples or definitions. In other words: Do what you like, or whatever the publisher's style guide requires.
Second: Technically, a counterexample is a proof of the negation of the assumption. A counterexample for "for every x and y there is a z such that xz = y" is a proof for "not for every x and y there is a z such that xz = y", or, if you push the negation inside, "there exist x and y such that for every z we have xz ≠ y".