We describe an improved nonequilibrium ionization (NEI) method that we have developed as an optional module for the FLASH magnetohydrodynamic simulation code. The method employs an eigenvalue approach rather than the earlier iterative ordinary differential equation approach to solve the stiff differential equations involved in NEI calculations. The new code also allows the atomic data to be easily updated from the AtomDB database. We compare both the updated atomic data and the methods separately. The new atomic data are shown to make a significant difference in some circumstances, although the general trends remain the same. Additionally, the new method also allows simultaneous calculation of the nonequilibrium radiative cooling, which is not included in the original method. The eigenvalue method improves the calculation efficiency overall with no loss of accuracy. We explore some common ways to present the NEI state with a sample simulation and find that using average ionic charge difference from the equilibrium tends to be the clearest method.