Step 1: Use Gauss's law directly.
Gauss's law states that the net electric flux through any closed surface equals the enclosed charge divided by \(\varepsilon_0\):
\[\Phi = \frac{q_{enclosed}}{\varepsilon_0}\]
Step 2: Charge inside the cylinder.
The field \(E\) is a uniform external field and the cylinder is non-conducting with no charge placed on or inside it. Hence \(q_{enclosed} = 0\).
Step 3: Apply.
\[\Phi = \frac{0}{\varepsilon_0} = 0\]
Step 4: Physical picture.
In a uniform field every field line that enters the cylinder through one end must come out through the other end. Since as many lines enter as leave, the incoming (negative) flux cancels the outgoing (positive) flux, and the curved wall contributes nothing because the field just grazes along it. The net flux is therefore zero.
\[\boxed{\Phi_{total} = 0}\]