As an extension of our previous work, we investigate the dynamical instability against non-axisymmetric bar-mode deformations of differentially rotating stars in Newtonian gravity by varying the equations of state and velocity profiles. We performed the numerical simulation and the follow-up linear stability analysis by adopting polytropic equations of state with polytropic indices n= 1, 3/2 and 5/2, and with two types of angular velocity profiles (the so-called j-constant-like and Kepler-like laws). It is confirmed that rotating stars with a high degree of differential rotation are dynamically unstable against bar-mode deformation, even when the ratio of the kinetic energy to the gravitational potential energy β is of order 0.01. The criterion for the onset of bar-mode dynamical instability depends weakly on the polytropic index n and the angular velocity profile, as long as the degree of differential rotation is high. Gravitational waves from the final non-axisymmetric quasi-stationary states are calculated using the quadrupole formula. For proto-neutron stars of mass 1.4 M⊙, radius ∼30 km and β≲ 0.1, such gravitational waves have a frequency of ∼600–1400 Hz, and the effective amplitude is larger than 10−22 at a distance of about 100 Mpc, irrespective of n and the angular velocity profile.