The claim is certainly true for $n$ sufficiently large, and "sufficiently large" could be specified explicitly with more care. 

We follow the suggestion of Fedor Petrov, and rely on the results of Brüdern & Fouvry (J. reine angew. Math. 454 (1994), 59-96) and of Heath-Brown & Tolev (J. reine angew. Math. 558 (2003), 159-224). These yield, for any prime $p\geq 5$, that if we count the number of representations $N=a_1^2+a_2^2+a_3^2+a_4^2$ smoothly (with positive integers $a_i$), then the proportion of representations such that $p\mid a_1$ is at most $1/(p-1)+o(1)$, where $o(1)$ is uniform in $p$ (here we use that we can restrict to $p\leq N^{1/2}$). See especially Theorem 3 and Lemma 6 of Brüdern & Fouvry, and the displays (340), (346), (349), (352) in Heath-Brown & Tolev. It follows that for $p\geq 7$ the proportion of representations such that $p\mid a_1a_2a_3a_4$ is at most $2/3+o(1)$, hence for $n$ sufficiently large (independent of $p$) there is a representation such that $p\nmid a_1a_2a_3a_4$. For $p=5$ we need to be more careful and subtract the (positive) proportion of representations such that two of the $a_i$'s are divisible by $p$. This is also covered by the mentioned works as they discuss general divisibility constraints $d_i\mid a_i$.