Given a list $a$ of integers, $$n_{a_1}, n_{a_2}, ..., n_{a_d}$$ have $$N_a = \sum_{j=1}^d n_{a_j}^2.$$ The various $N_a$, $N_b$ etc. are integers, but are not contiguous: for example, if $d=2$, there's way to obtain $N=7$.
Are the $N$ for a fixed $d$ countably infinite? Does it hold for any $d$?
If I understand your question correctly, for any fixed $d$, you are considering the set $S_d$ of all integers that can be expressed as a sum of $d$ squares (of integers).
For $d = 2$, there is Fermat's theorem on sums of two squares which leads to the consequence that (see here) any integer $n$ can be expressed as the sum of two squares if and only if every prime divisor of the form $4k+3$ occurs to an even power.
For $d = 3$, there is a theorem which states that a number $n$ can be expressed as the sum of three squares if and only if it is not of the form $4^k \left({8 m + 7}\right)$ for some $k$ and $m$.
For $d \ge 4$, there is Lagrange's four-square theorem which says that any integer can be expressed as a sum of $d$ squares.
In all cases (for all $d$), the set of $N$ that can be expressed as a sum of $d$ squares is a countably infinite subset of $\mathbb{Z}_{\ge 0}$ (the nonnegative integers), and is in fact identical to $\mathbb{Z}_{\ge 0}$ if $d \ge 4$.