I got this answer from Dusa McDuff (and she got it from some body else):
Suppose given $f:[0,1]\to [0,1]$ such thqt 0 is repelling fixed point and 1 is attracting fixed point and there are no others.
So $f'(0) = \lambda >1$, and $f'(1)=\mu < 1$.
A thm says that in suitable local coords near $0$ $f$ is simply mult by $\lambda$ (this is a linearization them). Therefore f has a unique square root on [0,1). Similarly, it has a unique square root on (0,1].
But in general the coords at the two ends will NOT be compatible
so there is no square root on [0,1].
Now consider a smooth $f: S^2\to S^2$ with two non-deg fixed points
$p_0,p_1$ with a
homoclinic orbit $A$ between them. i.e. there is an arc $A$ which at one end is the unstable manifold of $p_0$ and at the other is the stable manifold of $p_1$. Now restrict f to A.
(There is a stable manifold thm that says that locally these invariant submanifodls exist etc.)