The incorporation of rubber recycled aggregates from end-of-life tyres (ELT) in the manufacturing process of sustainable building materials has gained great interest in recent decades as a result of the large volume of this waste being generated annually. In this work, the objective is to make a contribution towards the circularity of construction products by carrying out a physico-mechanical characterisation of new gypsum composites made with the incorporation of these recycled rubber aggregates. To this end, up to 30% by volume of the original raw material has been substituted, analysing the mechanical resistance to bending and compression. Although lower than those of traditional gypsum material, both properties exceed the limits set at 1 and 2 MPa, respectively, by the current regulations. In addition, water absorption by capillarity significantly decreases, and thermal conductivity is reduced by more than 35% with respect to the reference material. Finally, in order to provide the research with a practical application, a prefabricated plate design has been proposed that incorporates the gypsum materials studied and an agglomerated rubber band that increases the thermal resistance and improves the efficiency of the designed construction system. In this way, this research reflects the potential of these novel building materials and explores new avenues for their application in building construction.