There are several fundamental ways to stop a noise like this. There are no magic bullets in the sense of being, easy, cheap and effective.
First way is to eliminate the source. If the whine is a piezo effect in the capacitors you have to change the capacitor type as others have already suggested. It is also possible that defects in physical construction could be allowing pure electrostatic attraction to be the transducer-again change the capacitors. Finally you can try to eliminate the ac component of the voltage on the capacitors which is driving the transduction mechanism. This is basically impossible in existing designs, and would drive things like selecting a linear architecture for supplies if you have control of the design. So not a very attractive solution.
The second way is to add damping to reduce the magnitude of the motion. Putting some lossy material where the whining component has to move it and allowing it to suck up the energy. The problem is that most glues and RTVs are not very lossy. Something like wet mud might work, but keeping it wet and where it belongs is a real problem. There are some audio materials that claim to be lossy. Some actually are, but nothing magic.
A third way is to shift the resonant frequency of the whining capacitor away from the driving frequency. This doesn't eliminate the noise, but can reduce the amplitude a lot. Changing stiffness is one way, and this is what glue attempts to do, but it often isn't stiff enough to really help. A rigid glue like epoxies or cyanoacrylate is the right direction, but may cause other problems due to CTE mismatches as temperature changes. Changing mass is the other way, and can be more effective. Lead foil or solder glued on with a rigid glue.
A fourth way is insulation. The component still whines, but the sound is confined and you don't hear it. Sound absorbing materials aren't that hard to find, but it is hard to find all of the paths by which the sound escapes and also hard to block some of them once identified. The best way to implement this method is to turn the equipment on and then leave the room. There are actually a lot of situations where this can work.