  Spontaneous phase separation instabilities with the formation of various
types of charge and spin pairing (pseudo)gaps in $U>0$ Hubbard model including
the {\it next nearest neighbor coupling} are calculated with the emphasis on
the two-dimensional (square) lattices generated by 8- and 10-site Betts unit
cells. The exact theory yields insights into the nature of quantum critical
points, continuous transitions, dramatic phase separation instabilities and
electron condensation in spatially inhomogeneous systems. The picture of
coupled anti-parallel (singlet) spins and paired charged holes suggests full
Bose condensation and coherent pairing in real space at zero temperature of
electrons complied with the Bose-Einstein statistics. Separate pairing of
charge and spin degrees at distinct condensation temperatures offers a new
route to superconductivity different from the BCS scenario. The conditions for
spin liquid behavior coexisting with unsaturated and saturated Nagaoka
ferromagnetism due to spin-charge separation are established. The phase
separation critical points and classical criticality found at zero and finite
temperatures resemble a number of inhomogeneous, coherent and incoherent
nanoscale phases seen near optimally doped high-$T_c$ cuprates, pnictides and
CMR nanomaterials.
