How does Boscovich's argument show that force must act at a distance? Apparently Boscovich's argument shows that force must act at a distance:

But how did the velocity of the faster body come to be reduced from 12
  to 9, and that of the slower body increased from 6 to 9? Clearly, the
  time interval for the change in velocities cannot be zero, for then,
  argued Boscovich, the instantaneous change in speed would violate the
  law of continuity. Furthermore, we would have to say that at the
  moment of impact, the speed of one body is simultaneously 12 and 9,
  which is patently absurd.
It is therefore necessary for the change in speed to take place in a
  small, yet finite, amount of time. But with this assumption, we arrive
  at yet another contradiction. Suppose, for example, that after a small
  interval of time, the speed of the faster body is 11, and that of the
  slower body is 7. But this would mean that they are not moving at the
  same velocity, and the front surface of the faster body would advance
  through the rear surface of the slower body, which is impossible
  because we have assumed that the bodies are impenetrable

Wouldn't be possible to explain this situation by discussing how the body is compressible instead?
 A: the problem with those arguments is that the description of collisions based solely on Newtonian mechanics is a good approximation that gives good results in terms of speeds before of after the collision, but if you try to force them beyond the intended use, they could lead to ridiculous (or contradictory) conclusions.
What actually happen in a real collision, is that the forces between the two bodies are electrical, they are the result of repulsion between the electrons of the atoms at the edge of both bodies. The electric force is a long range distance, so the interaction between the two colliding bodies starts quite before they enter in contact (actually they never enter into physical contact, unless there is a chemical reaction, because the repulsive forces will become strong enough to stop the bodies from actually "touching", I mean, if you had a very powerful zoom, you'll see that they never physically touch. this of course happen in a very small space and time scale, and macroscopically it is ok to idealize and assume that they actually touch each other to describe the results of an experiment. But to use that simplification to make arguments about what actually happens at the microscale, you will end up with those contradictions 
