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Assumed, that the ZF is consistent, you are right, of course. The fact, that there are rational points inside every cube, and that those cubes have no common volume, only common surfaces (lines, points) perhaps - is enough to state that there are at most aleph-zero of them.
But, if the ZF is not consistent, there may be a way to encode every real number between 0 and 1 with some peculiar cubic subdivision of 3D space.
You can easily encode every real between 0 and 1 via dividing (even a finite volume of) 3D space into disjunct 2D circles.
Perhaps, just perhaps, it is possible to replace those circles with cubes and spheres, given the infinite volume you have.
Perhaps, just perhaps, the ZF is broken.
People have attacked the consistency of ZF with much more powerful tools and failed. Your attack wouldn't sound promising to these people, because it already sounds unpromising to me. Each further minute of your time spent on this problem would be a failure of rationality.
If you want another problem that's really fun, prove or disprove that every division of the square into triangles of equal area will have an even number of triangles :-)