"such that" logical symbol

You could write this in a few different ways... I'm not sure what you're asking, so let me show you a couple.

For one, you could define the condition $y\in\text{Sqrt}(x)$, rather than the set itself: $$ y\in\text{Sqrt}(x)\Leftrightarrow y^2=x $$

The following two are commonly used in set definitions: $$ \text{Sqrt}(x)=\{y\mid y^2=x\}\qquad \text{or}\qquad \text{Sqrt}(x)=\{y:\ y^2=x\} $$

I also see people use (and have used myself) "s.t." as an abbreviation for such that in formulas.

Usually, there doesn't need to be a symbol other than a colon or $\mid$ for "such that."

The English language version of your statement seems to describe $\sqrt x$ as a set. You could write this as:

$$y\in \sqrt{x} \iff y\in\mathbb R \land y\cdot y = x$$

Note, I've added the $y\in\mathbb R$ because you need to know the domain in which you are working. You could chaange that, of course.

This is often abbreviated as:

$$\sqrt{x} =\{y\in\mathbb R\mid y\cdot y = x\}$$

Roughly, the $\mid$ character functions as a "such that" symbol here. Sometimes a $:$ symbol is used instead.

I think I remember that I have seen notations such as $$ \sqrt x :=\iota y (y\ge 0\land y^2=x)$$ i.e. $\iota v \Phi$ is used to denote the unique element of the (hopefully) singleton set $\{v\mid \Phi\}$. While having such a notation may be useful for extreme formality, I am personally no friend of it.