In a static fluid compressive force per unit area is called pressure. First off let me clear a misunderstanding that you have viz. "...a compressive force on one plane can be a shear force on another plane if that plane is oriented at right angle to the original plane". This is simply incorrect. Stress is not a vector but a tensor, and pressure is simply the isotropic part of that tensor. Now you must be aware that a tensor is a linear map from vectors to vectors, which is to say that a tensor takes vector as an input and outputs another vector. Now stress tensor $\tau$, by definition, takes area vector as its input, and outputs force vector (per unit area) acting on that area. Therefore to find force on any oriented plane located at $\mathbf{x}$ and whose unit normal is $\mathbf{n}$, you must take the stress tensor at that point $\tau_\mathbf{x}$ and feed $\mathbf{n}$ into it to get the force-per-unit-area vector $\mathbf{f}=\tau_\mathbf{x}(\mathbf{n})$. The force vector on a plane with a different orientation, say $\mathbf{n}'$, located at the same point, is $\mathbf{f}'=\tau_\mathbf{x}(\mathbf{n}')$, and in general $\mathbf{f}'\neq \mathbf{f}$. In static fluid, the force vector (per unit area) is always normal to any plane, and therefore if you take two orthogonal planes the corresponding force vectors will also be orthogonal. That is why compressive force on one plane does not become shear force on another plane; the forces on the two planes are to be computed separately and there is no obvious relation between them.
Now coming to your primary question: We can show that compressive stresses must exist in a static fluid by descending to the molecular level and using the definition of stress as momentum flux per unit area across an oriented plane due to molecules which are perpetually in motion. But I think you are seeking an explanation at the continuum level.
In a restricted sense, you "... could as well say there is no compressive force in a static fluid...". The reason is that usually (though not always) pressure differences are all that matter, and therefore you may consider any value of pressure to be your zero reference (just like in the case of potential energy). Pressure measured with respect to some such (arbitrary) zero reference is called "gauge pressure". But I think what you have in mind is "absolute pressure", in which the vaccuum is taken to be your reference for measurement of pressure in any other system.
Whenever external forces act on a body it will cause stress inside the body (there could be mean motion/rotation as well, but that does not concern us). In an open container of fluid, even neglecting contribution due to gravity, there is the atmosphere pushing down on the fluid and container walls that are pushing from the sides. What if we go into deep space where there is no gravity and just consider a free blob of liquid not inside any container? There is still surface tension force to reckon. If the fluid blob becomes too big (e.g. stars) there are forces due to self gravitation.
So to summarize:
- How do we prove/show that there are compressive forces in a static fluid?
When a body is acted upon by external forces it sets up stresses inside the body. If those stresses cannot be shear (because by definition, shear stresses induce motion in a fluid), then they must be compressive stresses.
- How do we prove/show that a fluid element does not move continuously under the influence of compressive forces?
Because presence of compressive stress does not imply presence of shear stress.
