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 $\mod \bmod p$. 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 for $k \neq 0$ we have $\gamma(k) = (k/p) \gamma(1)$ ([where $(\cdot/p)$ is the Legendre symbol)symbol], and $$p \gamma(1) = \sum_{a\bmod sum_{a \bmod p} e^{2\pi \exp(2\pi i a^2/p}a^2/p),$$ 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 each value of the convolution approaches $1/p$ at the exponential rate $p^{-n/2}$, and the probability you asked for approaches $(p-1)/(2p)$ at the exponential same rate$p^{-n/2}$..
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 $p^{-n/2}$.