Question: The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic equation.

But does it work in the opposite direction?

If for example for a certain matrix $A$ we know that

$ A^2-6A+9I=0, $

does that mean that the characteristic equation of $A$ is

$
\lambda^2-6\lambda+9=0
$ 
?

Answer: 
Even without counterexamples it is obvious that your statement can't be true because if $A$ is a root of the polynomial $p(x)$ then it must be the root of $p(x)q(x)$ for any polynomial $q$. So that way we would get the matrix $A$ has infinitely many characteristic polynomials.
For any $n \times n$ matrix $A$, the $(2n) \times (2n)$ matrix $\pmatrix{A & 0\cr 0 & A\cr}$ satisfies the characteristic polynomial of $A$, but its own characteristic polynomial is the square of that of $A$.