LSB of the Exponent in the DL Problem Can Be Efficiently Computed for Groups of Even Order
Write $x=x_0+2x_1$ for $x_0\in\left\{0,1\right\}$, so that the goal becomes determining $x_0$. Now compute $y^n$ (in terms of $h$, $x_0$ and $x_1$), and determine its order. What happens when $x_0=0$ and when $x_0=1$? Can we easily differentiate between them?

Note that this is the idea underlying the Pohlig-Hellman algorithm, so it may help to have a look at that.