Low-temperature electrical and thermal transport, magnetic penetration depth, and heat capacity measurements were performed on single crystals of the actinide superconductor UTe₂ (T_c=1.6K) to determine the structure of the superconducting energy gap. Heat transport measurements performed with currents directed along both crystallographic a- and b-axes reveal a vanishingly small residual fermionic component of the thermal conductivity. The magnetic field dependence of the residual term follows a quasi-linear increase consistent with the presence of nodal quasiparticles, rising rapidly toward the a-axis upper critical field where the Wiedemann-Franz (WF) law is recovered. Together with a quadratic temperature dependence of the magnetic penetration depth up to T/T_c= 0.3, these measurements provide evidence for an unconventional spin-triplet superconducting order parameter with point nodes positioned along the crystallographic a-axis. Millikelvin specific heat measurements reveal an upturn below 300 mK that is well described by a divergent quantum-critical contribution to the density of states (DOS). Modeling this contribution with a T^-1/3 power law allows restoration of the full entropy balance in the superconducting state and reveals a perfect T³ power law for the electronic DOS below T_c which is consistent with the point nodal gap structure determined by thermal conductivity and penetration depth measurements.