How can $N(x|\mu, \sigma^2)$ not be 0? 
How can $N(x|\mu,\sigma^2)$ not be 0? Because a Gaussian distribution is continuous and therefore there are an infinite number of values that x can occupy in the Gaussian distribution therefore the probability of any point should be 0. So I feel like the idea that you can calculate the conditional probability of a point given the parameters contradicts the idea that the probability of any value in the range of $N()$ must be 0, because it's continuous.

Every Gaussian distribution has so-called support over the entire real line. This means that the density function that you wrote never touches the x-axis.
A random variable $X$ with a Gaussian distribution has zero probability of taking by a value of zero: $P(X=0)=0$.
These are different notions, however similar they seem. For instance, a uniform $U(-a,a)$ distribution is zero on most of the real line, except for the interval $(-a,a)$, yet a random variable $Y$ with a $U(-a,a)$ distribution also has $P(Y=0)=0$.
One property of a continuous distribution is that every point has zero probability density.
