Proof by contradiction using counterexample Why can't we use one counterexample as the contradiction to the contradicting statement?
Example:
Let a statement be A where a-->b.
We can prove A is not true by finding a counter example.
Now, in another space and time, Let a new statement be B where it is the same as a-->not b.
Why can't we prove B is not true by finding a counter example?
Thus, a contradiction and A is true?

You can disprove the statement $a \implies \sim b$ using a counterexample if you can find one.  However, this does not prove the statement $a \implies b$.  This is because the negation of $a \implies \sim b$ isn't $a \implies b$.
Recall that the negation of an implication is not an implication.  $\sim (a \implies b) \equiv a \wedge \sim b$.  In words, this says that the negation of "a implies b" is equivalent to "a and not b".
You are right that if a statement $B$ is the negation of a statement $A$, then $A$ being false implies $B$ is true, and $B$ being false implies $A$ is true.  The problem is, the negation of an implication is not an implication.
