Suppose we extend the $[n,k]$ linear code $C$ over the field $\Bbb F_q$ to the code $C'$, where
$$ C' = \{(x_1,\ldots ,x_n,x_{n+1})\in \Bbb F_q^{n+1} : (x_1,\ldots,x_n) \in C \text{ and } x_1^2+\ldots+x_{n+1}^2 = 0\} $$
Under what conditions is $C'$ linear?
In general, the extended code $C'$ is linear if and only if the map $C \to \mathbb F_q$, mapping a codeword to its extension symbol is linear.
In your case, the extension may or may not be linear, depending on the original code $C$. The problem is that squaring is not linear, normally.
However, there is a large class of cases where we can directly answer the question:
If $q$ is even, then squaring is an automorphism of $\mathbb F_q$ (Frobenius automorphism) and thus, the extension rule is indeed linear. So whenever $q$ is even, $C'$ is linear.