Yes, any generator will be subject to this effect.
You can't get something for nothing, so the more current drawn from the coils, the larger the (opposing) magnetic field (counter mmf) and the harder the rotor will be to turn.
If this didn't happen then the amount of energy required to turn the rotor would not change according to the load on the output, which would break the laws of physics.
If you have a 100W electrical load and you drive 100W plus efficiency losses, say 110W, into the generator, things will be in a state of equilibrium, with 100W being converted from mechanical input power into electricity, and the other 10W of mechanical input power being eaten up by losses.
Now suddenly put 1kW of mechanical power into the machine; at that instant, before the rotational speed can change, the 100W electrical load will continue to present the same mechanical load to the prime mover. Things will not be in equilibrium, and the machine's rotational speed will accelerate. Depending on circumstances, this may or may not increase the electrical load. Certainly the generated voltage will go up, and any simple resistive load will therefor absorb more power, but maybe you have some regulation such that the load continues to draw exactly 100W.
So assume the load continues to draw exactly 100W. Where does the extra 900W of mechanical power go then? The machine's speed must increase until the losses equal the mechanical driving power; so it ends up turning extremely fast, the increased power going into increases in friction in the bearings, windage loss due to the rotating parts, eddy currents in the magnetics (and doubtless a couple of other things I forget at the moment), none of which are desirable.
You would find that, without exceeding the machine's electrical rating, you would quickly exceed its mechanical ratings, i.e., probably long before you got to 1000W, the rotation speed would be several times the suggested speed, and catastrophic failure would likely result. Note you can do this with no electrical load on the generator at all.
Best Answer
Some will, some won't.
Most self respecting engines have electronic motor management these days, to regulate output voltage and frequency. Often you see really decent machines built into a sea container, extremely silent and the only way to tell that a generator is in them is because you're all of a sudden walking through a beam of hot air. These containers work as a Faraday cage, which in turn will probably make them insusceptible again.
The cheap and tiny ones are often consumer stuff, they can't afford electronic management and are unstable in both voltage and frequency aswell as depending on load characteristic.
On the other hand ... it all depends on the energy contained in the EMP ... I guess you can never be really sure. Don't know if EMP is viable to melt generator windings ...