I know that a beam suspended in two ends can have many modal shapes. However, when it comes to unsupported free beam its first three eigenfrequencies are 0, why? And why should it have eigenfrequencies at all if it is free to move? Is eigenfrequency characteristic of every material? Why is it important?
The zero frequencies are the rigid body translations and rotations of the structure. (If you think there are 3 rather than 6, presumably you are only considering motion in two dimensions not three). The non-zero frequencies represent vibration modes where the beam bends. There does not have to be any fixed restraints for bending to happen. Imagine a beam pinned at the ends. Then replace suspend the beam on two ropes. Then replace the ropes with flexible cords. Then take away the cords completely (and ignore gravity!). You can imagine a continuous progression from the pinned beam to a completely free beam, and there is no reason why vibrations would suddenly become impossible. Real-world examples of "unsupported" structures that can vibrate (and where understanding the vibrations are important) are aircraft, rockets, spacecraft, etc.