The equality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are unequal non-zero constants, is satisfied by $x=am+bn$, where:
$\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad \\ \textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad \\ \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad \\ \textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad \\ \textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value}$
Expand binomials, combine like terms, and subtract terms from both sides.
\[x^2 + 2xm + m^2 - x^2 - 2xn - n^2 = m^2 - 2mn + n^2\]
\[2xm + m^2 - 2xn - n^2 = m^2 - 2mn + n^2\]
\[2xm - 2xn - n^2 = -2mn + n^2\]
Get all the x-terms on one side and factor to solve for x.
\[2xm - 2xn = -2mn + 2n^2\]
\[2x(m-n) = -2n(m-n)\]
Since $m \not= n$, both sides can be divided by $m-n$.
\[x = -n\]
That means $a = 0$ and $b = -1$, so the answer is $\boxed{\textbf{(A)}}$.