Multicomponent quantum chemistry allows the quantum mechanical treatment of electrons and specified protons on the same level. Typically the goal is to identify a self-consistent-field (SCF) solution that is the global minimum associated with the molecular orbital coefficients of the underlying Hartree-Fock (HF) or density functional theory (DFT) calculation. To determine whether the solution is a minimum or a saddle point, herein we derive the stability conditions for multicomponent HF and DFT in the nuclear-electronic orbital (NEO) framework. The gradient is always zero for an SCF solution, whereas the Hessian must be positive semi-definite for the solution to be a minimum rather than a saddle point. The stability matrices for NEO-HF and NEO-DFT have the same matrix structures, which are identical to the working matrices of their corresponding linear response time-dependent theories (NEO-TDHF and NEO-TDDFT) but with a different metric. A negative eigenvalue of the stability matrix is a necessary but not sufficient condition for the corresponding NEO-TDHF or NEO-TDDFT working equation to have an imaginary eigenvalue solution. Electron-proton systems could potentially exhibit three types of instabilities: electronic, protonic, and electron-proton vibronic instabilities. The internal and external stabilities for theories with different constraints on the spin and spatial orbitals can be analyzed. This stability analysis is a useful tool for characterizing SCF solutions and is helpful when searching for lower-energy solutions. Initial applications to HCN, HNC, and 2-cyanomalonaldehyde, in conjunction with NEO ∆SCF calculations, highlight possible connections between stationary points in nuclear coordinate space for conventional electronic structure calculations and stationary points in orbital space for NEO calculations.