Defining $N$ in the $\epsilon$-$N$ definiton of convergence

Here's the real reason why $N$ has to be a natural number: a real-valued sequence $(s_n)$ is a function $\mathbb{N}\to\mathbb{R}$, where for notational convenience we write $s_n$ to represent $s(n)$. In the $\varepsilon$-$N$ definition of the limit of a sequence everything with $\varepsilon$ has to do with the outputs of this function, which are real numbers; but everything with $N$ and $n$ describes inputs of this function, which are natural numbers. So from the point of view of the domain of this function, non-natural real numbers don't even exist: there's no such thing as "$s(0.5)=s_{0.5}$" in this sequence. So letting $N$ to be real would be easier… but it wouldn't be meaningful.

Your professor is correct in saying that ultimately it doesn't really matter: saying that $n\ge123.456$ is as good as saying $n\ge124$. But I believe that this is a bad style, because the spirit of the definition — and even its letter, as stated in your book! — require $N$ to be natural.

Moreover, once the definition has been stated, it's better to stay consistent with it. So you're also right in your observation that it's very confusing and very sloppy to state one definition and then do something different. Especially, since there's such an easy fix: say, in the example in your post simply let $N=\left\lceil1/\sqrt{\varepsilon}\right\rceil$.

$N$ is often required to be a natural number in the definition since it is used to say something about all integers $n\geq N$. Since we have a sequence indexed by the naturals, it is "nicer" to think about all integers $n$ greater than or equal to, say $4$, rather than all integers $n$ greater than or equal to $\pi$. Both sets, though, are the same, which is why your professor said the definitions are equivalent.