First of all, is the ring $\left(\mathbb{Z}, 0, 1, +, *\right)$ more or less the same as $\left(\mathbb{N}, 0, 1, +, *\right)$, except the negative integers? If so, I'm most of the way there:
We have that, for an arbitrary n-tuple $\left(w_0,...,w_n\right) \in \mathbb{N}$, $\beta(a, d, 0) = w_0,...,\beta(a, d, t)=w_t$. Since d=s!, where s = $max\left(t+1, w_0,...,w_t\right)$, we have that all $\left(d_0,...,d_t\right)$ are relatively prime, so $\left(\mathbb{N}, 0, 1, +, *\right)$ can be coded, since the $\beta$ function is clearly arithmetical.
For the integers, I am stuck. We're told that each integer can be written as the sum of four squares but, I don't see how to write $\left(w_0,...,w_n\right)$ as $\left[\left(u^2_0+v^2_0+s^2_0+t^2_0\right),...,\left(u^2_t+v^2_t+s^2_t+t^2_t\right)\right]$ - as far as I can tell, no sum of four squares can be less than 0.
Finally, why is the integer structure called a ring while the natural number structure is not?
Well the key is to realize that you can identify the natural numbers$ \def\nn{\mathbb{N}} \def\zz{\mathbb{Z}} $ within the ring of integers $\zz$ in the language of rings. Namely, you can find a $1$-parameter sentence $P$ such that $\zz \vDash P(n)$ for every $n \in \zz_{\ge 0}$ but $\zz \vDash \neg P(n)$ for every $n \in \zz_{<0}$. This is possible by the 4-square theorem. Once you have this you can easily translate anything expressible using arithmetic over $\nn$ to arithmetic over $\zz$, including Godel-coding. Namely, "$\forall n\in\nn\ ( \cdots )$" translates to "$\forall n\in\zz\ ( P(n) \to \cdots )$" and it should be clear how to do it for "$\exists$" as well.