The general form of the covariance depends on the first three moments of the distribution. To facilitate our analysis, we suppose that $X$ has mean $\mu$, variance $\sigma^2$ and skewness $\gamma$. The covariance of interest exists if $\gamma < \infty$ and does not exist otherwise. Using the relationship between the raw moments and the cumulants, you have the general expression:
$$\begin{equation} \begin{aligned}
\mathbb{C}(X,X^2)
&= \mathbb{E}(X^3) - \mathbb{E}(X) \mathbb{E}(X^2) \\[6pt]
&= ( \mu^3 + 3 \mu \sigma^2 + \gamma \sigma^3 ) - \mu ( \mu^2 + \sigma^2 ) \\[6pt]
&= 2 \mu \sigma^2 + \gamma \sigma^3. \\[6pt]
\end{aligned} \end{equation}$$
The special case for an unskewed distribution with zero mean (e.g., the centred normal distribution) occurs when $\mu = 0$ and $\gamma = 0$, which gives zero covariance. Note that the absence of covariance occurs for any unskewed centred distribution, though independence holds for certain particular distributions.
Extension to correlation: If we further assume that $X$ has finite kurtosis $\kappa$ then using this variance result it can be shown that:
$$\mathbb{V}(X) = \sigma^2
\quad \quad \quad \quad \quad
\mathbb{V}(X^2) = 4 \mu^2 \sigma^2 + 4 \mu \gamma \sigma^3 + (\kappa-1) \sigma^4.$$
It then follows that:
$$\begin{align}
\mathbb{Corr}(X,X^2)
&= \frac{\mathbb{Cov}(X,X^2)}{\sqrt{\mathbb{V}(X) \mathbb{V}(X^2)}} \\[6pt]
&= \frac{2 \mu \sigma^2 + \gamma \sigma^3}{\sqrt{\sigma^2 \cdot (4 \mu^2 \sigma^2 + 4 \mu \gamma \sigma^3 + (\kappa-1) \sigma^4)}} \\[6pt]
&= \frac{2 \mu + \gamma \sigma}{\sqrt{4 \mu^2 + 4 \mu \gamma \sigma + (\kappa-1) \sigma^2}}. \\[6pt]
\end{align}$$
For the special case of a random variable with zero mean we have $\mu=0$ which then gives:
$$\mathbb{Corr}(X,X^2)
= \frac{\gamma}{\sqrt{\kappa-1}},$$
which is the scale-adjusted skewness parameter.
