Can we squeeze atoms? Lets consider few water molecules (or monoatomic noble gas or $H^+$ ions or electron gas) to be present inside a vessel, now if we enclose the vessel with certain hard material which could act as piston (assume that vessel is completely covered). Now, if we apply tremendous force on the piston (and no molecules are rushing out of the covering), can we watch the molecules to be squeezing?    
I have heard about particle accelerators, where we start with electrons or protons, accelerate  them to high energy and smash into the target. In general, heavier the particle we want to produce, the higher must be energy of collision.   
It seems that keeping the high energy particles stationary and then squeezing them (if possible) with highly accelerated pistons from either side, is better than colliding high energy particles head-on.   
The problem in not using pistons, I must think, must be because of our inability to accelerate the massive pistons to a high extent as like accelerating particles itself. But, the head-on collision of pistons with high energy particles in between, will be better than head-on collision of particles itself, I think.   
In case of head on collision of particles I think, we can accelerate them to high extent, but we can't expect effective collision.  
Balancing both mass and acceleration of piston, can we have the same result, as we get in particle accelerators? If we can, can we avoid constructing those large particle accelerators like Stanford linear accelerator center (see the picture)?    
Can we create such piston squeezer? I know there might be some difficulties? What can we do to overcome it? 

Sometimes, I might have misunderstood, if so, pardon me and explain.
 A: 
Can we squeeze atoms?

Yes. High pressure changes the wavefunctions of electrons in atoms.  See for example Accurate Wavefuctions for the Confined Hydrogen Atom at High Pressures.
One effect of this is to increase the rate of electron capture by the nucleus, since s electrons will spend more time at/in the nucleus at high pressure.
See the lecture: Pressure and Chemistry Dependent Electron Capture for more information.
You definitely will not reach the energy levels of an accelerator however, as others have explained.  
A: The experiments at LHC hit protons on each other at total energy of 7 TeV.
In comparison, a flying moscquito has a kinetic energy of 

1 TeV: a trillion electronvolts, or 1.602×10−7 J, about the kinetic energy of a flying mosquito[12]

Could one collide two flying bees  of 3.5TeV kinetic energy and get a proton proton collision? i.e. give the total energy of the bee on a proton of each bee's proboscis?
The answer is "no" because macroscopic objects as the bee's proboscis have a surface composed by a very large number of atoms and the contact of the two proboscises will disperse the energy among those atoms much before it can reach one proton.
So it is true that macroscopic objects have a lot of kinetic energy in TeV, but to get it on an elementary particle level is a problem that has not been solved. Actually, maybe, with nanotechnology something might be devised, with magnetic fields and electric fields and God knows what, but certainly it will not be with pistons.
A: Degenerate matter is essentially squezed atoms.
It's not man-made,
But it is perfectly squeezed atoms, all of them.
And real squezed in the sense of static pressure.
In cores of stars, the pressure get's so high - only after the end of hydrogen burning - that the electrons can no longer stay withe the nuclei.
See Degenerate matter  and Electron degeneracy pressure
(That can even happen further regarding various subatomic particles,squeezing even what was left from the atoms)
A: Once you get up to relativistic speeds, there's no such thing as a "hard" material.  If you could get a pair of macroscopic pistons to collide at the location of a test atom, the dynamics of the collision would be dominated by collisions between the atoms in the pistons.
You can collide a light particle with a heavy nucleus — this is what's done at RHIC and LHC in the proton-gold and proton-lead collisions.  And you can collide two heavy nuclei — gold-gold, or lead-lead.  But the heavy-nucleus collisions are quite messy.  Even if you could make the beams dense enough to make a gold-proton-gold collision (which you couldn't), a gold nucleus has 197 nucleons in it; it'd be pretty tough to distinguish between a gold-gold collision involving 394 nucleons and a gold-proton-gold collision involving 395 nucleons.
A: The short answer would be not really (at least the way you've described it). The problem with any sort of physical piston is that high energy particles vibrate and will collide with the piston. This would have several undesirable effects. Firstly, it could damage the piston depending on how its made. Secondly, the collision would heat up the piston causing the particles to lose energy.
However, it is possible to use magnetic fields or lasers to confine particles. These paricles can be confined at high energies (essentially squeezed as you put it). This is the idea behind nuclear fusion. Note fusion particle energies tend to be on the order of $keV$ compared to $7~TeV$ at the LHC so it can't really tell you the same physics.
An alternative idea of squeezing atons would be a Bose-Einstien condensate or similar. This is almost the complete opposite in that the gas is cooled to very near absolute zero. In this case all the particles occupy the lowest possible quantum state so in one sense they are as squeezed as possible. 
