Let $X_1, \cdots, X_n$ be $n$ iid random variables such that each $X_i$ follows from the continuous uniform distribution in $[-1,1]$.
I am looking for a lower bound on $$ \mathbb{E}\left[\sqrt{\sum_{i=1}^{n}X_i^2}\right]. $$
Note that by using Jensen's inequality, we can provide an upperbound:
$$ \mathbb{E}\left[\sqrt{\sum_{i=1}^{n}X_i^2}\right] \leq \sqrt{\sum_{i=1}^{n}\mathbb{E}\left[X_i^2\right]} = \frac{\sqrt{n}}{\sqrt{3}}, $$ where the last step follows from simple calculation.
Based on @VazenBu comment, we can provide the following lowerbound.
Let $z_1,\dots, z_n$ be $n$ numbers such that $z_i \in [0,1]$. Then, we can show that $$ \frac{\sum_{i=1}^{n}z_i}{\sqrt{n}} \leq \sqrt{\sum_{i=1}^{n}z_i}. $$
We can use the above inequality to argue that
$$ \frac{\mathbb{E}\left[\sum_{i=1}^{n}X_i^2\right]}{\sqrt{n}} \leq \mathbb{E}\left[\sqrt{\sum_{i=1}^{n}X_i^2}\right]. $$ Note that $\mathbb{E}\left[X_i^2\right]=\frac{1}{3}$. Therefore, we have $$ \frac{\sqrt{n}}{3} \leq \mathbb{E}\left[\sqrt{\sum_{i=1}^{n}X_i^2}\right]. $$