The minimum of $J$ is not unique $m=2$, $n=3$, $E=\{(1,2),(1,3),(2,3)\}$ and $X=\mathbb R$.

Assume that the points $a_1$ and $a_2$ are far enough from each other. Then at the minimum of
$$d^2(a_1,u_1)+d^2(a_2,u_2)+2{\cdot}d(u_1,u_2)$$ the points $u_1\ne u_2$ lie at certain places of geodesic $[a_1a_2]$.

Now, $$J(u)= d^2(a_1,u_1)+d^2(a_2,u_2)+d(u_1,u_2)+d(u_1,u_3)+d(u_3,u_2)\ge\\ \ge d^2(a_1,u_1)+d^2(a_2,u_2)+2{\cdot}d(u_1,u_2)$$ and equality holds if $u_3\in[u_1u_2]$. So, you should take $u_1$ and $u_2$ as above and any $u_3\in[u_1u_2]$.

The minimum of $J$ is not unique $m=3$, $n=4$, $E=\{(1,2),(1,3),(2,4),(3,4)\}$ and $X=\mathbb R^2$.

Assume that $a_1$, $a_2$ and $a_3$ are vertices of huge equilateral triangle. Then at the minimum of
$$d^2(a_1,u_1)+d^2(a_2,u_2)+d^2(a_3,u_3)+\\+d(u_1,u_2)+d(u_1,u_3)+d(u_2,u_3)$$ the points $u_i$ lie are vertices of slighlty smaller equilateral triangle insde $\triangle a_1a_2a_3$.

Now, $$J(u)= \\ =d^2(a_1,u_1)+d^2(a_2,u_2)+d^2(a_3,u_3)+\\+d(u_1,u_2)+d(u_1,u_3)+d(u_2,u_4)+d(u_3,u_4)\ge\\ \ge d^2(a_1,u_1)+d^2(a_2,u_2)+d^2(a_3,u_3)+\\+d(u_1,u_2)+d(u_1,u_3)+d(u_2,u_3)$$ and equality holds if $u_4\in[u_2u_3]$. So, you should take $u_1$, $u_2$ and $u_3$ as above and any $u_4\in[u_2u_3]$.