Quantifying a free variable in an example from "How To Prove It" by Velleman This is an example from How To Prove It by Daniel J. Velleman (2nd Ed., p. 71):

Example 2.2.3. Analyze the logical forms of the following statements.
  
  
*
  
*Statements about the natural numbers. The universe of discourse is $\mathbb N$.
a. $x$ is a perfect square.

And here is the solution:

  
*
  
*a. This means that $x$ is the square of some natural number, or in other words $\exists y(x = y^2)$.
  

My question is regarding the free variable $x$. Is it correct to leave it free, as the author has done? I was under the impression that, by convention, a free variable may be assumed to be universally quantified, which I think wouldn't make sense in this case.
If I wanted to fully formalize the statement, how would I do that? My best guess is as follows:
$$\exists x\exists y(x = y^2)$$ 
Is this correct?

You shouldn't confuse variables (the only ones that can be quantified) with individual constants. 
The logical form of the sentence "$x$ is a perfect square" is indeed $\exists y (x = y^2)$. In this formula, $x$ plays the role of an individual constant, not of a variable, because it refers to one specific individual in the universe of discourse $\mathbb{N}$. 
The meaning of $\exists x \exists y (x = y^2)$ is "there is a perfect square", which is completely different because it does not refer to one specific individual in the universe of discourse $\mathbb{N}$.

Remark. The starting point of my answer above is the fact that the exercise claims that "$x$ is a perfect square" is a statement, that is a a meaningful declarative sentence that is true or false because every sign in it is interpreted, hence $x$ should refer to a specific individual in the universe of discourse. 
If you see "$x$ is a perfect square" just as a phrase which is not a sentence (and so it is not true or false because not all its signs are interpreted), then you can see $x$ as a free variable (waiting for an interpretation to assign a truth value to the phrase). Even in this case the logical (but meaningless) form of the phrase is $\exists y (x = y^2)$.
