In both white dwarfs and neutron stars, collapse occurs because hydrostatic equilibrium cannot be achieved by decreasing the radius. This process occurs even if the electrons or neutrons remain present and unchanged, but might be accelerated if some process removes them.
Details
White dwarfs and Newtonian gravity
Here is a simple Newtonian argument for why a collapse must eventually occur in white dwarfs of increasing mass, governed by relativistic electron degeneracy pressure, even if no neutronisation occurs.
As the mass of a white dwarf increases, its radius decreases and its density increases. At low densities, the degenerate electrons have non-relativistic energies, and the pressure $P \propto \rho^{5/3}$. As the density increases, the degenerate electrons get pushed to relativistic energies and the relativistic degeneracy pressure goes as $P \propto \rho^{4/3}$.
What supports a star against gravity is the pressure gradient, $dP/dr$, through the equation of hydrostatic equilibrium
$$ \frac{dP}{dr} = - \rho g\ . $$
If we just deal in proportionalities so that $dP/dr \propto P/R$ and $\rho \propto M/R^3$, then in the low density, non-relativistic case, with $g \propto M/R^2$, we have
$$\frac{M^{5/3}}{R^6} = \frac{M^2}{R^5}\ .$$
For any given increase in mass it is then possible to decrease $R$ to keep the LHS equal the increasing RHS.
In the relativistic case, the hydrostatic equilibrium equation gives
$$\frac{M^{4/3}}{R^5} = \frac{M^2}{R^5}\ . $$
In this case, the equation can only work for a single mass. Any increase in the mass above that means the RHS would become bigger than the LHS and the star will collapse. This limiting mass is the Chandrasekhar mass.
It is worth noting then that the pressure provided by fermion degeneracy will always be "overcome" at some finite mass threshold even in Newtonian physics. Consideration of General Relativity (see below) simply lowers the mass threshold.
In practice, the "real" Chandrasekhar mass is a little lower in typical carbon white dwarfs because indeed, electrons are captured by protons in the nuclei once they become highly relativistic; this removes electrons and lowers degeneracy pressure, leading to collapse.
Neutron stars and General Relativity
In neutron stars, the reason for the upper limit is also because hydrostatic equilibrium cannot be reached, either because of the increasing density (even if the neutrons remain intact) but possibly accelerated if the neutrons are removed.
We cannot use the Newtonian hydrostatic equilibrium equation in neutron stars. Its General Relativistic equivalent is the TOV hydrostatic equilibrium equation.
$$\frac{dP}{dr}=-\left(P+\rho\right)\frac{m(r)+4\pi r^3P }{r\left(r-2m(r)\right)}\ .$$
A major difference is that the pressure appears on the RHS. This means that increasing the density at the centre of the star in order to increase the pressure gradient and support a more massive star also increases the pressure gradient required to support that star. Ultimately this is self-defeating and the RHS will always be bigger than the LHS, for any radius, and the star collapses. The mass threshold for collapse is lower than it would be if Newtonian hydrostatic equilibrium were considered.
There is considerable debate however as to whether this process might be accelerated by the disappearance of neutrons. This might happen because they have enough energy to create heavy hadrons - the hyperons - like $\Sigma$ and $\Lambda$ particles. This would have the effect of turning neutron kinetic energy, which is a source of pressure, into additional rest-mass energy and thus decreases pressure for a given density and might destabilise the star.
More catastrophic may be the production of mesons via strong force interactions - pions or kaons. These feel the strong nuclear force but are bosons, can form a condensate, and so that component of the pressure contributed by neutron (fermion) degeneracy would be removed and might trigger the collapse.
There are other possibilities too - like quark matter, but it is unclear whether that would hinder any collapse. The discovery of neutron stars of mass $2M_\odot$ possibly means that equilibrium structures featuring quark matter do not exist (e.g. Ozel & Freire 2016), but I'm sure you can also find papers that disagree.