We present a study of the one-dimensional attractively interacting two-component Fermi gas with an external magnetic field. We analytically study the quantum phase diagram, equation of state, quantum criticality, and universal scalings of the model, using the thermodynamic Bethe ansatz equations. We present a high precision equation of state from which universal scaling functions are derived; we show that the thermodynamic properties display universal scaling behaviour at quantum criticality. Expression of high-order-accuracy entropy and specific heat are also analytically derived. Moreover, we present an independent confirmation of the equation of state through an efficient numerical approach of treating the thermodynamic Bethe ansatz equations for which the contributions from the spin strings with length larger than a value of n_c are calculated analytically.