Well-defined, solid continuum materials generally have the same compressive and tensile Young's moduli.
The reason is that the elastic moduli measure slight displacements from an energy minimum, typically representing the equilibrium spacing of atoms balanced between attraction and repulsion. All smooth energy minima look like symmetric parabolas close up. Thus, the restoring forces from symmetric slight compression and tension are nearly identical, and so are the corresponding Young's moduli.
What about bone? Bone is a biological material with a variety of compositions and some degree of porosity. The description "bone" is too ambiguous: What bone, from what animal? Was the same material used for both the compressive and tensile tests? How did the researchers correct for the discrepancy between the measured cross-sectional area and the actual material area, given the porosity? And in particular, did the compressive displacement involve slight atomic shifting only, or did some of the struts forming the porous cells buckle, an effect that wouldn't occur in tension? (Differences in compressive and tensile behavior in such porous materials is discussed in Gibson's & Ashby's Cellular Materials.)
For these reasons, I'd expect nearly identical compressive and tensile Young's moduli in most engineering materials (solid ceramics, metals, polymers below the glass transition temperature) but am not very surprised to see different experimental results for an entry labeled simply "bone." I would agree with you (in contrast to the other answer, which I think deeply confuses stiffness and strength) that bone is a special case in this context.