My bars move side-to-side when I take my hands off
So, don't take your hands off the bars!
If you take your hands off the handlebars on your bicycle or your car's steering wheel, you'll notice that you continue going straight. If you do this with the Ninja, you'll many times notice the bars start oscillating to the point of violence and wanting to chuck you off the bike.
Now, if you've ever taken your hands off your car's steering wheel while going in reverse, you'll see that the wheel will turn hard-over. Your motorcycle, because of the engine braking pressure applied from the rear tire while you're in gear, is acting the same way (among other forces that keep the bike from just going hard to one side and flopping over, hence the oscillations). The physics of this relate to the caster angle and the direction of force. Since a bicycle is in neutral, and other motorcycles aren't wound up so high and putting such a degree of engine braking in by the combination of high rpms and low weight, we Ninja riders are the ones who really notice it, and it always surprises us.
Think of the gyroscopic motion of the wheels as being a type of spring (or a tight violin string). There is a preferred direction, and when you perturb it, it wants to go back to it, but it can overshoot and then pull back again, setting up the oscillation.
The wheelbase length is also something to think about. The shorter the wheelbase, the more "closely coupled" the gyroscopic forces of the front and rear tires will be. That, combined with the lightness (low mass therefore low inertia) of the EX-250, makes for a much more delicately balanced "center neutral" situation.