Can neural networks approximate any function given enough hidden neurons?

The key point to understand is compact:

Neural networks (as any other approximation structure like, polynomials, splines, or Radial Basis Functions) can approximate any continuous function only within a compact set.

In other words the theory states that, given:

1. A continuous function f(x),
2. A finite range for the input x, [a,b], and
3. A desired approximation accuracy ε>0,

then there exists a neural network that approximates f(x) with an approximation error less than ε, everywhere within [a,b].

Regarding your example of f(x) = x², yes you can approximate it with a neural network within any finite range: [-1,1], [0, 1000], etc. To visualise this, imagine that you approximate f(x) within [-1,1] with a Step Function. Can you do it on paper? Note that if you make the steps narrow enough you can achieve any desired accuracy. The way neural networks approximate f(x) is not much different than this.

But again, there is no neural network (or any other approximation structure) with a finite number of parameters that can approximate f(x) = x² for all x in [-∞, +∞].