The multiferroic properties of EuTiO3 are greatly enhanced when a sample is strained, signifying that coupling between strain and structural, magnetic or ferroelectric order parameters is extremely important. Here resonant ultrasound spectroscopy has been used to investigate strain coupling effects, as well as possible additional phase transitions, through their influence on elastic and anelastic relaxations that occur as a function of temperature between 2 and 300 K and with applied magnetic field up to 14 T. Antiferromagnetic ordering is accompanied by acoustic loss and softening, and a weak magnetoelastic effect is also associated with the change in magnetization direction below ~2.8 K. Changes in loss due to the influence of magnetic field suggest the existence of magnetic defects which couple with strain and may play a role in pinning of ferroelastic twin walls.