Interpreting the statement Just started quantifiers, and I'm having some trouble with interpreting this.

Here's what I understand:

For every Epsilon $&gt; 0$, there exists a Delta $&gt; 0$ for all $x$ in $\mathbb{R}$.

The antecedent of the implication is false, because there is no Delta that is always greater than $x$.

The precedent of the implication is also false, because not every Epsilon greater than $0$ is also greater than $x^2 -1$ for any $x$.

So the statement is false. Is my logic sound? I'm not sure if I'm reading this right.
I believe your difficulty arises from a shortage of parentheses in the formula.  I think the formula was intended to be 
$$
(\forall\varepsilon&gt;0)(\exists\delta&gt;0)(\forall x\in\mathbb R)\Big([0&lt;|x-1|&lt;\delta]\implies [|x^2-1|&lt;\varepsilon]\Big).
$$
That is, the quantifiers at the beginning apply to the whole implication, not just to its antecedent.  This is also how the two answers preceding mine (by Ahmed and Michael Cotton) interpreted the formula.
You could read the implication as:

$|x^2 -1|&lt;\varepsilon\:$ whenever $\:
|x-1|&lt;\delta$

which seems just fine.

With the quantifiers, this whole thing just says the function $x^2$ is continuous at $x=1$.

Added note: See Andreas Blass' answer. I was assuming the quantifiers applied to the entire implication. Depending on the text it's from, that's probably what is intended.