Expectation of $\langle s,x \rangle^2$
Let us assume that each entry of $s$ is drawn independently from some distribution $\mathcal{D}$ whose expectation is $0$. Then
$$
\mathbb{E}\left[\left(\sum_{i=1}^d s_i x_i\right)^2\right] =
\sum_{i=1}^d x_i^2 \mathbb{E}[s_i^2] + \sum_{i \neq j} x_i x_j \mathbb{E}[s_i s_j] = \|x\|^2 \mathbb{V}[\mathcal{D}],
$$
since $\mathbb{E}[s_is_j] = \mathbb{E}[s_i]\mathbb{E}[s_j] = 0$ due to independence.

In your particular case, $\mathcal{D}$ is the uniform distribution over $[-1,1]$, whose variance is
$$
\frac{1}{2} \int_{-1}^1 x^2 \, \mathrm{d}x = \left. \frac{x^3}{6} \right|_{-1}^1 = \frac{1}{3}.
$$