We report the results of resonantly enhanced, degenerate four-wave mixing (DFWM) measurements of the cubic molecular hyperpolarizability coefficient (gamma) of four squaraine dyes. Different determinations of gamma were performed at 696 and 710 nm with both ultrashort (210 fs) and somewhat longer (3 ps) duration laser pulses using the forward wave geometry. Bis[4-(dimethylamino)phenyl]squaraine (H-Sq), bis[4-(dimethy lamino)-2-hydroxyphenyl]squaraine (HO-Sq), bis(2,4,6-trihydroxyphenyl)squaraine (3HO-Sq), and diazulenylsquaraine (Az-Sq) were investigated as a function of concentration in liquid solutions and polymer blends, enabling the calculation of gamma from the measured values of the third-order nonlinear susceptibilities, chi((3)), of these samples. All values of gamma for these squaraines were found to be negative with large absolute magnitudes, up to similar to 8 x 10(-32) in esu, "units" [9.9 X 10(-57) C m(4) V-3 (SI units)] for the real part of gamma for Az-Sq at 696 nm. [We discuss the conversion of esu units to SI units for the quadratic (beta) and cubic hyperpolarizability coefficients (gamma) in the Appendix of this paper, reporting our results throughout in both systems of units.] Significant imaginary contributions to gamma were also observed for Az-Sq, but not for the other squaraines as expected from differences in absorption at these wavelengths. The DFWM signals exhibited the expected cubic intensity dependence, polarization dependence, and instantaneous time response, effectively ruling out any contributions to the measured values of gamma from experimental artifacts.