I wonder whether the following property holds true: For every real symmetric matrix $S$, which is positive in both senses: $$\forall x\in{\mathbb R}^n,\,x^TSx\ge0,\qquad\forall 1\le i,j\le n,\,s_{ij}\ge0,$$ then $\sqrt S$ (the unique square root among positive semi-definite symmetric matrices) is positive in both senses too. In other words, it is entrywise non-negative.

At least, this is true if $n=2$. By continuity of $S\mapsto\sqrt S$, we may assume that $S$ is positive definite. Denoting $$\sqrt S=\begin{pmatrix} a & b \\ b & c \end{pmatrix},$$ we do have $a,c>0$. Because $s_{12}=b(a+c)$ is $\ge0$, we infer $b\ge0$.