Compressively strained epitaxial (001) EuTiO₃ thin films of tetragonal symmetry have been deposited on (001) (LaAlO₃)_0.29-(SrAl_1/2Ta_1/2O₃)_0.71 (LSAT) substrates by reactive molecular-beam epitaxy. Enhancement of the Néel temperature by 1 K with 0.9%compressive strain was revealed. The polar phonons of the films have been investigated as a function of temperature and magnetic field by means of infrared reflectance spectroscopy. All three in-plane polarized infrared active phonons show strongly stiffened frequencies compared to bulk EuTiO₃ in accordance with first-principles calculations. The phonon frequencies exhibit gradual softening on cooling, leading to an increase in static permittivity. Additional polar phonon with frequency near the TO1 soft mode was detected below 150 K. This mode coupled with the TO1 mode was assigned as the optical phonon from the Brillouin zone edge, which is activated in infrared spectra due to an antiferrodistortive phase transition and due to simultaneous presence of polar and/or magnetic nanoclusters. In the antiferromagnetic phase, we have observed a remarkable softening of the lowest-frequency polar phonon under an applied magnetic field, which qualitatively agrees with first-principles calculations. This demonstrates the strong spin-phonon coupling in EuTiO₃, which is responsible for the pronounced dependence of its static permittivity on magnetic field in the antiferromagnetic phase.