Hookie's Law defines the linear elastic relationship of material within the elastic range in the stress-strain curve.
$F = kx$
$k = EA/L$, so
$F = EA(x/L) = EA\epsilon$, and $F/A = \delta$, so
$\delta = E\epsilon$
Now we can generalize the "Hooke's Law" to cover the stress ($\pm\delta$), on an element of "homogeneous isotropic" material, resulting from any types of loading that produces a strain ($\epsilon$) in direct proportion to the intensity of the stress (linear), and the strain is recoverable upon unloading (elastic behavior) in the elastic range.
Therefore, "Hooke's Law" holds for any material, posses homogeneous and isotropic properties, under any types of loadings or load combinations, that behave in a linear elastic manner.
The flaw in your thinking is "My guess is that Hooke's law is defined for an infinitesimal cubic element....". In fact, regardless of "size", a material/element exhibits linear stress-strain behavior (linear proportionality of stress and deformation) upon any type of loadings with the stressed plane remains plane, and upon unloading, it returns back to its original shape, is said it is a linear elastic material the Hooke's Law defined for.
ADD (11/2) - Influence of Staged Loadings with Consideration of Stiffness Variations (For Discussion Only)