Concrete cracking is a significant issue in the global construction industry, and the restraint stress of concrete is a crucial contributing factor to early concrete cracking. The addition of magnesium oxide additive (MEA) to concrete is a method to enhance its crack resistance. In this paper, concrete specimens with four different contents of MEA were tested with a temperature stress testing machine. The deformation characteristics and mechanical properties of concrete with varying contents of MEA were investigated using both free deformation tests and fully constrained deformation tests. The prediction model for the early restrained stress of concrete was developed by integrating the stress relaxation phenomenon of concrete with models for autogenous shrinkage, temperature deformation, and elastic modulus. According to the results, (1) the thermal expansion coefficient exhibits a pattern of initially increasing and subsequently decreasing with the increasing ratio of MEA; (2) the addition of 3% and 8% MEA can offset 23% and 35.1% of the concrete's self-shrinkage, respectively. Nevertheless, when the added MEA content is 5%, the self-shrinkage of concrete increases by 6%; (3) the addition of 3-8% MEA can result in a 0.5-1.67 times increase in the maximum expansion stress of concrete, as well as a 0.5-0.95 times increase in cracking stress; (4) as the MEA content continues to increase, the stress relaxation level of concrete also increases. In comparison to concrete mixed without MEA, the maximum increase in the stress relaxation level of concrete is 65.5%, thereby enhancing the concrete's anti-cracking ability. However, when the MEA dosage reaches a certain threshold, the stress relaxation enhancement brought about by the addition of MEA will no longer be significant; (5) when compared to the experimental data, the established model of early-age constraint stress accurately predicts the tensile constraint stress of concrete.