I think that appropriate modifications to the eunicycle could make it even easier to ride than the segway. While this is extremely counterintuitive, it makes a lot of sense from a physics standpoint. A car is completely based on static balance, meaning that if it turns off it doesn't roll over. A bicycle has some dynamic balance, because it would fall to the right or left if the rider stopped balancing. A unicycle is completely dynamic balance, which makes it much more difficult to ride, but also makes it far more controllable and maneuverable once you've learned to ride it. The way one shifts one's weight on a unicycle, and by extension a eunicycle, is ironically more intuitive for humans than the way it works on a bicycle, because it's the same as the way we do it with the form of locomotion we inexplicably use all the time, which is bipedalism. Bipedalism is completely based on dynamic balance, and requires constant corrections even when just standing still.
In order to make a eunicycle easier to ride, it must be made self-balancing. On a unicycle, and the eunicycle as it is today, you balance by waving your hands around, which is both difficult to do and very limited in the amount of force it can exert, which makes going at very high speed inherently dangerous since you can't maneuver. A more stable vehicle can be constructed as follows: On the bottom, there's a wheel like in the existing eunicycle. Above that, there's a horizontal flywheel attached to a high-torque motor which is used for balancing. Above that, and attached to the wheel via structural components which go around the flywheel, is a platform which the rider stands on, and in front of the rider there are handlebars of the same design as the segway for the rider to hold onto.
If the rider is moving forwards and leans right, the eunicycle turns the flywheel to the left, thereby turning the rider and wheel to the right and keeping the rider from falling over (although generally this will result in the vehicle being angles slightly more forward, so it will then accelerate to keep from falling forwards). Likewise, if the rider is going forward and leans left the flywheel is used to turn the rider and wheel to the left. If the rider is going backwards then the wheel is turned left to compensate for leaning right and right to compensate for leaning left.
A weird problem is that if you keep turning in the same direction for a while the flywheel might build up considerable angular momentum, eventually getting to the point where it can't turn any faster and hence the steering bottoms out. I'm not sure if this would be a real problem in practice, there are several ways the effect could be dampened or avoided if it does.
In principle this sort of vehicle should be able to go faster than a motorcycle, since it has only one wheel and hence half the friction, although in practice the weight of the flywheel and stability issues might limit its speed.
If you really wanted the ultimate high-speed vehicle, it would probably be a glycerine jet-propelled unicycle with electronic stabilization system, although that would be incapable of idling and be in some ways closer to a jet pack than a land vehicle.