This paper proposes a Joint Channel Estimation and Symbol Detection (JED) scheme for Multiple-Input Multiple-Output (MIMO) wireless communication systems. Our proposed method for JED using Alternating Direction Method of Multipliers (JED-ADMM) and its model-based neural network version JED using Unfolded ADMM (JED-U-ADMM) markedly improve the symbol detection performance over JED using Alternating Minimization (JED-AM) for a range of MIMO antenna configurations. Both proposed algorithms exploit the non-smooth constraint, that occurs as a result of the Quadrature Amplitude Modulation (QAM) data symbols, to effectively improve the performance using the ADMM iterations. The proposed unfolded network JED-U-ADMM consists of a few trainable parameters and requires a small training set. We show the efficacy of the proposed methods for both uncorrelated and correlated MIMO channels. For certain configurations, the gain in SNR for a desired BER of $10^{-2}$ for the proposed JED-ADMM and JED-U-ADMM is upto $4$ dB and is also accompanied by a significant reduction in computational complexity of upto $75\%$, depending on the MIMO configuration, as compared to the complexity of JED-AM.