Quadratic Hamiltonians that exhibit single-particle quantum chaos are called quantum-chaotic quadratic Hamiltonians. One of their hallmarks is single-particle eigenstate thermalization introduced in Łydżba et al. [Phys. Rev. B 104, 214203 (2021)2469-995010.1103/PhysRevB.104.214203], which describes statistical properties of matrix elements of observables in single-particle eigenstates. However, the latter has been studied only in quantum-chaotic quadratic Hamiltonians that obey the U(1) symmetry. Here, we focus on quantum-chaotic quadratic Hamiltonians that break the U(1) symmetry and, hence, their "single-particle" eigenstates are actually single-quasiparticle excitations introduced on the top of a many-body state. We study their wave functions and matrix elements of one-body observables, for which we introduce the notion of single-quasiparticle eigenstate thermalization. Focusing on spinless fermion Hamiltonians in three dimensions with local hopping, pairing, and on-site disorder, we also study the properties of disorder-induced near zero modes, which give rise to a sharp peak in the density of states at zero energy. Finally, we numerically show equilibration of observables in many-body eigenstates after a quantum quench. We argue that the latter is a consequence of single-quasiparticle eigenstate thermalization, in analogy to the U(1) symmetric case from Łydżba et al. [Phys. Rev. Lett. 131, 060401 (2023)0031-900710.1103/PhysRevLett.131.060401].