This probability can be calculated exactly, and indeed it approaches $1/2$ rather quickly —  more precisely, for each $p$ it approaches the fraction $(p-1)/(2p)$ of quadratic residues $\mod p$.  This can be proved by elementary means, but perhaps the nicest way to think about it is that if you choose $n$ numbers $a_i$ independently and sum $a_i^2 \bmod p$, the resulting distribution is the $n$-th convolution power of the distribution of a random single square — so its discrete Fourier transform is the $n$-th power of the D.F.T., call it $\gamma$, of the distribution of $a^2 \bmod p$.  For this purpose $\gamma$ is normalized so $\gamma(0)=1$.  Then $\gamma(k) = (k/p) \gamma(1)$ (where $(\cdot/p)$ is the Legendre symbol), and $p \gamma(1) = \sum_{a\bmod p} e^{2\pi i a^2/p}$, which is a Gauss sum and is thus a square root of $\pm p$.  It follows that $|\gamma(k)| = p^{-1/2}$, from which we soon see that the probability you asked for approaches $(p-1)/(2p)$ at the exponential rate $p^{-n/2}$.

As noted above, this result, and indeed the exact probability, can be obtained by elementary terms, yielding a (known but not well-known) alternative proof of Quadratic Reciprocity(!). But that's probably too far afield for the present purpose.