It is shown that the pseudo Jahn-Teller effect (PJTE) in combination with ab initio calculations explains the origin of instability of the planar configuration of tetrafluorocyclobutadiene, C(4)F(4), with respect to a puckered structure and square-to-rectangle distortion of the carbon ring, and rationalizes its difference from the planar-rectangular geometry of C(4)H(4) and nonplanar (puckered) structure of Si(4)H(4). The two types of instability and distortion of the high-symmetry D(4h) configuration in these systems emerge from the PJT coupling of the ground B(2g) state with the excited A(1g) term producing instability along the b(2g) coordinate (elongation of the carbon or silicon square ring), and with the excited E(g) term resulting in e(g) (puckering) distortion. A rhombic distortion b(1g) of the ring is also possible due to the coupling between excited A(1g) and B(1g) terms. For C(4)F(4), ab initio calculations of the energy profiles allowed us to evaluate the PJTE constants and to show that the two instabilities, square-to-tetragonal b(2g) and puckering e(g) coexist, thus explaining the origin of the observed geometry of this system in the ground state. The preferred cis-trans (e(g) type) puckering in C(4)F(4) versus trans-trans puckering (b(2u) distortion) in Si(4)H(4) follows from the differences in the energy gaps to their excited electronic E(g) and A(1u) terms causing different PJTE in these two cases.