Q: Do box wings suffer from induced drag the same way as normal wings?
A: Yes and no. Box Wing aircraft will suffer from induced drag the same as any aircraft will, if they are heavier-than-air vehicles and are using their wings to fly. Induced drag is a function of finite span loading, and moderated by various ways to improve design efficiency at a given span loading. Thus the amount of drag, and the way it is created and avoided, differs for a boxwing and a monoplane of the same span. Today this topic of induced drag includes completely different definitions than what was taught in seminal references on the subject. Even if one is talking about the same thing, the topic will hear arguments from two different camps: those who adhere to representative mathematics, and those who focus on the non-Cartesian, non-textbook actual physics on a case-by-case basis. It's quite fair to say that the former are more vocally opinionated than the latter, for the latter know less until later.
The job of a wing is to efficiently push and pull air downward as it moves forward. That action causes both a Newtonian reaction and a Bernoulli pressure differential, resulting in lift.
Making lift this way causes the nearby air to also be affected, as a time-dependent secondary result. It has to 'fall into the descending trough of air' that the wings displaced downward.
This secondary movement causes (completely unavoidable) rotational movements in the "wake" zone between air directly moved by the wings and the nearby stationary air, thereby involving more air mass than the plane needed to move just to get the lift it needed. (The momentum difference is quite literally the induced drag, although we usually teach it in ways more related to how induced drag is visualized and computed in 2-D. Other answers posted here illustrate this in conventional terms.)
Induced drag and wake vortex CANNOT be eliminated for a lifting wing system of any kind. However, most aircraft wing designs allow something else to happen that greatly increases this cost of making lift with a finite wingspan: they let high pressures under the wing be 'too close' to the low pressures above the wing for the amount of pressure difference that has developed in flight. If a high differential pressure exists at a wing tip, a strong, tornado-like vortex will form there.
Allowing any strong gradient to form between low pressure and high pressure will cause air to move toward the low pressure at a high velocity, if it can. Drag increases exponentially with the velocities imparted to the air, therefore designers use a variety of approaches to keep this equalization from happening quickly. The slower it happens, the less kinetic energy is imparted to the air by the airplane.
This is where Boxwings have a totally different way of reducing the induced drag, compared to a normal wing: they put a wall up between the low pressure above the wing, and the higher pressure everywhere else. The 'wall' can be taller than a winglet, because it has a wing above to help resist the forces that push on it from the side. At that upper wing connection, the wall-like vertical surface of a boxwing likewise stands between the higher pressure under the wing, and the lower pressure everywhere else.
If a designer does a good job with this idea (many do not), both the biplane wing surfaces and the vertical surfaces of the boxwing system will moderate the velocity of gradient-induced airflows by acting against the undesirable flows in 3-D space. They become more effective in this with greater vertical spacing.
The easier and more effective way to reduce the induced drag is simply to increase the wingspan, or reduce the vehicle weight. As a wing gets longer, the portion of the lift each unit of the wing needs to make is reduced, meaning that it will have a lower pressure differential between the upper and lower surfaces. Best practice calls for this differential to be minimized at the tip, so the gradient is weakened. The result then is that a weaker pressure gradient and a longer distance between low and high pressures will keep the equalization velocities down.
However, as an aircraft gets heavier or goes faster, this approach becomes first very expensive, then impossible. Material strength limitations put definite limits on the wingspan of conventional aircraft.
Surprisingly, box wings fare no better... perhaps worse. What appears to be a structural advantage actually merely concentrates the bending forces, generated by each wing, into the corners of the box. Making them strong enough quickly becomes excessively heavy. Therefore, a box wing aircraft should, like a biplane, have a shorter span than a monoplane of equivalent induced drag. Its span efficiency bears greater fruit among short span designs, than where wingspan can be increased.
One might think that this advantage would then bear fruit indirectly, through speed. The faster an aircraft flies, for a given span loading, the less induced drag it will make. In fact, at high indicated airspeeds, induced drag becomes a small component of total drag. However, other aspects of box wing designs seem to have impeded high-speed boxwing solutions; notably stability; and "interference drag."
In a box wing design, there is a forward set of lifting wings, and an aft set of lifting wings. In high speed flight this configuration cannot respond as stably or as quickly to certain conditions as a wing with a (downward-lifting) tail.
When set up as a tandem-lifting wing arrangement without such a stabilizer, as is typical of modern versions, boxwings have to balance at their combined center of upward lift, rather than ahead of it like conventional aircraft do, thanks to the stabilizing influence of a tail pushing in the opposite direction. This limitation and tandem-wing stall behaviors place challenging, inherent demands on boxwing designs that constrain their success at higher flight speeds.
As noted above, they also create interference drag. This type of drag can be hard to predict and is also widely misunderstood. In practice, the inherent, 3-D interference drag of a boxwing aircraft design greatly reduces the 2-D theoretical advantage of the configuration toward obtaining induced drag benefits. This is why they are not at all like "normal wings."
As mentioned in the original post, there is a new aircraft configuration that is often mistaken for a box wing design. However, it is nothing like them. It's called a box-tail or double boxtail configuration. I am the designer of the Synergy double boxtail aircraft, which is the first such aircraft to be developed.
These somewhat disappointing attributes of the otherwise logical box wing configuration were at the heart of matters during the long period of Synergy's development. It was my desire to utilize high span efficiency and laminar flow in a high speed aircraft design, while avoiding high speed landings and unpredictable, unstable behaviors at low speeds. A video of a 25% scale model in flight and a basic overview can be seen at synergyaircraft.com. A post on the topic of boxwings can be found there as well.
For more information regarding span efficiency and non-planar configurations, Ilan Kroo has published very thorough overviews of the subject. The graphic below is adapted from one appearing in his papers. It shows how induced drag can be fought in 3-D space by moving away from a flat, planar wing into the vertical dimension. Synergy builds that understanding further, into the longitudinal and time dimensions, in accord with the concepts advanced first by George C. Greene while at NASA Langley.