Elgamal with secret key equal to zero possible?
It's worse than that: if $x = 1283767$, then the ciphertext will be $(g^r, m \cdot g^{r \cdot 1283767})$, which an adversary can immediately detect by checking whether $h = g^{1283767}$ and subsequently exploit to decrypt messages!  The same trick works for $x = -1$, $x = 42$, and $x = 123456789$.

We pick $q$ and $g$ so that the space of values for $x$ is so large that if we choose $x$ uniformly at random, the adversary's probability of success at guessing it is negligible, below $2^{-128}$—so small that you should be more worried about cosmic rays flipping bits in your RAM than you are about this event.

There are much better attacks recovering $x$ from $h = g^x$, requiring you to choose $q$ to have thousands of bits.  You should be much more worried about how you're using Elgamal encryption and how you shoehorn arbitrary messages in your application into integers mod $q$, and even more worried about articulating higher-level application security goals.