Linear and nonlinear oscillatory and extensional flow was studied for polymer layered silicate nanocomposites of organically modified bentonite in ethylene-vinyl acetate copolymer with 18 wt.% of vinyl acetate (EVA18). It was found that the rheological properties of EVA18 nanocomposites were distinctly different from the pure copolymer The elastic response at low frequencies was significantly enhanced in comparison to that of pure EVA18. The linear to nonlinear transition for stress relaxation measurements and the damping function were examined. The relaxation spectrum was shifted toward the longer relaxation time scales for EVA18 nanocomposites, while the dependence of the damping for EVA18 nanocomposites was much stronger than that of the EVA18. In contrast, the uniaxial extensional viscosity of EVA18 nanocomposites gave weaker strain-hardening properties. The prediction of strain-hardening behaviour for EVA18 and EVA18 nanocomposites through relaxation spectrum and the damping function based on Kaye-Bernstein-Kearsley-Zapas (K-BKZ) model is discussed. A simplified estimation of the nonlinear material parameter P in the K-BKZ model is proposed to predict more accurately the extensional viscosity for EVA18 and EVA18 nanocomposites. Experimental data and detailed predictions are also presented in this paper.