Is $X\sim(\mathcal{N}(0,1))^2$ allowed? Am I allowed to say that $X\sim(\mathcal{N}(0,1))^2$? Or should I instead say $X=Y^2$ where $Y\sim\mathcal{N}(0,1)$?
My question arises from the following: $\sigma\mathcal{N}(0,1)=\mathcal{N}(0,\sigma^2)$. 
I am not really sure what kind of mathematical object $\mathcal{N}(0,1)$ is: am I allowed to take its square? Should I write $\sigma\mathcal{N}(0,1)\sim\mathcal{N}(0,\sigma^2)$ instead? 
 A: The notation $X\sim \mathcal{D}$ means "The random variable $X$ has law (is distributed according to) the probability distribution $\mathcal{D}$.
$\mathcal{N}(0,1)$ is standard notation for the standard normal distribution over $\mathbb{R}$, i.e. the continuous real-valued probability distribution with probability density function $f\colon x\mapsto \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$.
Writing $X\sim\mathcal{N}(0,1)$ thus makes sense. Writing $\sigma\mathcal{N}(0,1)=\mathcal{N}(0,\sigma^2)$, for $\sigma>0$, also makes sense (as a common notation), since $\mathcal{N}(0,\sigma^2)$ is the Gaussian/Normal distribution with mean zero and variance $\sigma^2$. (Note that this is again a notation: you shouldn't think of it as the usual multiplication, more as a common shortcut for "$\sigma\mathcal{N}(0,1)$ is the distribution of $\sigma X$, when $X\sim\mathcal{N}(0,1)$." The key is that the notation is common and quite standard.)
Writing $x\sim(\mathcal{N}(0,1))^2$, however, does not (unless you want to use it as an ad hoc notation, in which case you need to define the new operation on probability distributions $\mathcal{D}\mapsto\mathcal{D}^2$ which maps a probability distribution to another legit probability distribution). What is $(\mathcal{N}(0,1))^2$? Is it a probability distribution, as it should? 

In your specific case of the square of a Gaussian random variable, anyway, I'd recommend using the standard notation $X\sim\chi^2$, since by definition this is what the chi-squared distribution is...
A: The notation $\mathcal{N}(\mu,\sigma^2)$ is used to refer to a random variable. So $X \sim \mathcal{N}(\mu,\sigma^2)$ should be read: "$X$ is Normally distributed with mean $\mu$ and variance $\sigma^2$". 
Though you can always invent notation if you find it useful, in this case it seem unhelpful to apply functions to the property itself. Instead the functions should be applied to the random variable which will then affect the property. 
For instance,
$$\text{if } X \sim \mathcal{N}(0,1),\;\text{ then }\;bX \sim \mathcal{N}(0,b^2).$$
Or, more generally
$$\text{if } X \sim \mathcal{N}(\mu,\sigma^2),\;\text{ then }\;a+bX \sim \mathcal{N}(a+b\mu,b^2\sigma^2).$$
