Definition of sequential continuity: converse?

Take any constant function. Then $y_n$ is trivially convergent for any $x_n$.

Probably not. Example: $f(x) = x^2$ is continuous on $\mathbb{R}$, and take $x_n = (-1)^n$ does not converge, but $y_n=f(x_n) = 1, \forall n$, hence converges to $1$. Another example would be to take $f(x) = |x|$, and the same $x_n = (-1)^n$ would work just fine.