I was excited to see this question because it harkened back to my childhood, and the glories of Bizarro World. But since you want science, let's consider the possibilities.
If you have artificial gravity, you can make the world any shape you want. Since gravity reshapes space itself, however, we could get into arguments about whether or not it is really cubical, or simply flat. From my perspective, I don't care: artificial gravity is the easy way out, so I'm going to ignore it.
As mentioned before, the relationship of gravity to the strength of your world is critical in defining the stability of any cubical world. How strong depends significantly on the size of your world, which (per the IAFA definition) cannot be a planet no matter its size:
"A celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit."
A cube cannot be "nearly round" by any definition I consider valid (though particle physicists may disagree. Moreover, the surface of your world will of necessity be boring.
Specifically, consider the gravity of your world near the edge of the cube. "Down" will still point towards the center of the cube (not with mathematical precision, but that's the limit as your approach the edge. The bigger your planet, the less the difference matters). But the surface of the planet near that same edge is lifted 45° from "down", which means that any liquid poured onto the surface of your world would flow towards the middle of any given side.
Moreover, the tendency of matter to liquefy under pressure means that the surface of your cubical world would either need to be sufficiently small that this process would not occur, or it would need to have a solid surface made of whatever material is strong enough to keep it square. Indeed, if you do the math, you'll find that any world big enough or hot enough to support a liquid mantle will be pressing up on the sides, and the forces are strongest in the middle of each face. Since most solids are stronger in compression than in tension, the tensile strength at the surface is probably the limiting requirement for your planet.
If you're willing to go with fictional materials, Larry Niven's scrith would do the job -- its tensile strength is roughly equal to the nuclear strong force. If you want a real material, however, you're going to have to work at it:
- Consider the asteroid/dwarf planet Ceres. It is roughly 945 km (587 miles) across, with a surface gravity of 0.27 m/s2. If our space engineers rework it into a cube, it would be approximately 760 km across.
- Let's assume that the force twisting our cube out of square would be roughly equal to the pressure exerted by (945 - 760)/2 = 92) km of missing mass at the center of each face. This isn't precisely true because surface gravity on a cube is different, but it's close enough IMHO for an envelope calculation.
- Pressures beneath the earth go up by 7300 psi per mile of depth, or 31.5GPa per km of depth. Even at Ceres' gravity (I'll assume it scales linearly), that's 542 KPa per km of depth. At 92 km deep, we're talking about a pressure of almost 50 GPa.
- The strongest known substance is graphene, with a theoretical maximum strength of 300 GPa. This is good - but your world probably isn't made of graphene, not if it's a natural satellite. Also, you'd have to be working with a pure carbon asteroid to make a pure graphene object. Also, since graphene is flexible, it would have to be structured into some kind of three-dimensional arrangement to keep the surface rigid.
- An alternative to consider is silicon carbide. It is tougher than graphene, and almost as strong under tension. Since you need a surface that not only won't break, but also won't bend, you may want to look into this. Whether or not it will break depends, I think, on how much of the planet is structure and how much is creamy filling, but at least you'll be able to use up any silicon dust you have lying around to make the structural part.
Note that even this case of extreme engineering still gives you a relatively small world, unable to hold any significant atmosphere, and it can't have surface water either. Artificial gravity, or post-transuranics, or some other unobtanium with previously unknown material characteristics may be required.
