Mixed strategies: Nash equilibrium
Hint 1: Suppose a mixed strategy NE $(\sigma_1,\sigma_2)$ exists, where $\sigma_i$ is a distribution over $[0,1]$. By the property of MSNE, 
$$u_i(x_i,\sigma_j)=u_i(x_i',\sigma_j)$$ 
for all pure strategies $x_i,x_i'$ in the support of $\sigma_i$. 

Hint 2: Note that a segment of $u_i$ is linear in strategies, and thus 
$$
x_i+\frac12(1-x_i-\mathbb E[x_j])=x_i'+\frac12(1-x_i'-\mathbb E[x_j])
$$