Answer
a. See proof.
b. No, the converse is not true.
Work Step by Step
a. Consider a cubical Gaussian surface of side s, with the uniform electric field perpendicular to a face. The flux through that face is $-Es^2$ and the flux through the opposite face is $Es^2$. The flux through the other 4 sides is zero because the electric field is perpendicular to the area vector. In summary, the net flux through this closed surface is zero. Gauss’s Law shows that the enclosed charge is zero.
b. Consider a bubble in empty space, then place a point charge just outside the bubble. The electric field will not be uniform inside the bubble, though it contains no charge.
