This is a very good question. Let's generalize the regularization approach that Sangchul presented.
Let $\delta_n(x)$ be a sequence of (at least piece-wise smooth) positive-valued functions such that
$$\lim_{n\to \infty}\delta_n(x)=\begin{cases}0&,x\ne 0\\\\\infty&,x=0\tag1\end{cases}$$
and for each $n$
$$\int_{-\infty}^\infty \delta_n(x)\,dx=1\tag2$$
and for all suitable test functions, $\phi(x)$, we have
$$\lim_{n\to \infty}\int_{-\infty}^\infty \delta_n(x)\phi(x)\,dx=\phi(0)\tag2$$
We say that in distribution $\delta(x) \sim \lim_{n\to \infty}\delta_n(x)$. This is the Dirac Delta distribution and $\delta_n(x)$ is a regularization thereof.
Now, let's examine whether $\lim_{n\to \infty}\delta_n(x^2)$ has meaning in the distributional sense. We can write
$$\begin{align}
\int_{-\infty}^\infty \delta_n(x^2)\phi(x)\,dx&=\int_0^\infty \delta_n(x)\left(\frac{\phi(\sqrt{x}\,)+\phi(-\sqrt{x}\,)}{2\sqrt{x}}\right)\,dx\\\\
&=\int_{-\infty}^\infty \delta_n(x)\left(\frac{\phi(\sqrt{|x|}\,)+\phi(-\sqrt{|x|}\,)}{2\sqrt{|x|}}\,H(x)\right)\,dx
\end{align}$$
where $H(x)$ is the Heaviside function. Letting $n\to \infty$, we find for suitable test functions (smooth and of compact support)
$$\lim_{n\to \infty}\int_{-\infty}^\infty \delta_n(x^2)\phi(x)\,dx=\begin{cases}0&,\phi(0)=0\\\\\infty&,\phi(0)>0\\\\-\infty &,\phi(0)<0\end{cases}$$
So, $\delta(x^2)$ has meaning in distribution on the space of $C^\infty_C$ functions that are $0$ at the origin and the distribution assigns the value of $0$ to each of these functions. As Sangchul wrote, this is quite a "boring" distribution.
Now, if $\phi(x)$ is a test function such that $\phi(0)=0$, smooth a.e. except at $0$ where $\lim_{x\to 0^{\pm}}\phi'(x)=C^{\pm}$, then we see that
$$\lim_{n\to \infty}\int_{-\infty}^\infty \delta_n(x^2)\phi(x)\,dx=\left(\frac{C^+-C^-}{2}\right)\,H(0)$$
Inasmuch as $H(0)$ is not uniquely defined, then we cannot uniquely define $\delta(x^2)$ as a distribution acting on such functions. For example, if $\phi(x)=|x|$, then $C^\pm=\pm1$ and $\delta(x^2)\sim H(0)$, which is not uniquely defined.