To connect the spheres you need to connect them dynamically with an nConstraint, not with bones.
Here is a scene that does what you describe:
In this case I created two cloth nodes. The squishy spheres started as cubes with poly smooth(to keep vertex count low), then poly combined then made into nCloth. The final out cloth mesh is then smoothed(3 key). To do the non-squishy ones I poly combined 3 planes such that the vertex positions were where I wanted the black spheres. Then I made that nCloth and set the nClothShape solverDisplay to collision thickness and then set the collision and self collision flags to vertex, and set the thickess to the desired size of the black spheres. (instead one could use nParticles for the spheres, but this was a simple way to create the correct positions)
To constrain the the spheres I selected both cloth objects and created a component to component constraint. Then on the constraint set connectionMethod to Within Max Distance and adjusted the max distance just high enough to create links. (you can use greater distance for more links and stiffness).
With the above you still need to go through hoops if you want to render the black spheres correctly. It is probably simpler just to make them like the squishy cloth but with higher stretch/compression/bend resistance… just keep the base model very low resolution (like a cube with one smooth) then smooth the output cloth mesh to make it look better. The max distance contraint will also make the small spheres very stiff if the max distance is higher than the radius of the small spheres( they will get lots of internal cross links then).