The probability of even a Hydrogen atom popping into existence is astronomically low, but the point is that one could. And if one can, then many can, and on and on. The point is that it's not impossible, just improbable. But by definition of one did pop into existence, it's a stable state and would therefore not just simply vanish after forming.
I think the time-energy uncertainty relation limits their distance and it is very small, closer to a Planck length than any precision we can measure. They have to be close enough to annihilate within a specific time limit related to the energy of the particles (ΔE·Δt ≥ ℏ/2) and the higher energy the particles like a full proton, the less time they have.
I think maybe the original Boltzmann brain may have been about pre-existing matter particles having the ability to just "jump" to a new location (similar to how they can tunnel thru a barrier), and so you can have them all just "jump" into the shape of a perfect square, perfect crystal, or even perfectly formed cat.
So it's not necessarily about the "from pure empty space" version of it. This is all just a thought experiment and not intended to be taken literally.
Honestly, you have a series of good questions and unfortunately I don't think your other interlocutor is helping.
I'll give it a try. This is very much a hand-wave.
tl;dr: Boltzman brains (Bb) from energy gradients: a "formless" high-energy fluctuation in a cold gas returns to thermal equilibrium through a Bb state.
The classical cartoon for a Standard Model of Particle Physics human Boltzmann brain (SMBb) is a cold (too cold for metabolism) Maxwell-Boltzmann gas or soup of atoms which in principle could condense into biological molecules via chains of endothermic and exothermic reactions. The gas is on average too cold for the endothermic parts, though, which essentially means no products comparable to those from anabolism.
From fluctuation theory randomly there will be high-temperature regions which allow for endothermy. In a hot bubble, which quickly cools by dissipation, complex molecules are (briefly) energetically favourable as their formation also lowers, locally, the temperature of the hot spot in which it's embedded.
The Boltzmann brain formed in this way is liable to be cooked apart by the heat of the hot spot outside the small region that cooled as the Bb condensed in it.
You've been wondering about how an SMBb might form. The starting point is a cold gas of photons. In that gas regions with vast amounts of gammas can fluctuate out of equilibrium: this is the hot spot. Particle pair productions and complicated decay chains serve to cool the hot spot (inelastically scattering gammas, among other processes), and consequently forming complex bound states embedded within the super-hot-but-rapidly-cooling spot is energetically favourable. Yes one has to have some luck (or some extra constraint or mechanism) in matter/antimatter asymmetry, which should be encoded in the grand canonical ensemble for interacting particles.
The SMBb will be torn apart by gammas and other radiation in the hot spot in which it is momentarily a cold spot.
One might compare this with a different extremely hot "hot spot" in which complicated states can form briefly. High-entropy pulsional pair-instability supernovae (PPISN) <https://online.kitp.ucsb.edu/online/stars-c17/woosley/pdf/Wo...> (see especially the graph on slide 5) are massive stars whose core rises to pair production temperatures. The pair-production cools the core of the very hot (~ 0.3 GK), very massive star; the cooling means less radiation supporting the star's bulk above the core. Gravitation from the sheer mass of the star itself drives a thermal runaway. However, during the runaway, there will be further cooling via nuclear processes that tend to generate a large neutrino flux. Soon however the star and all the heavy daughter products which condense during the pair-producing phase tend to be violently explosively disassembled.
The "hot spot" for a Standard Model Boltzmann brain might be several terakelvins hotter than a PPISN's core (the SMBb bubble ought to be gamma gas reaching QCD or even GUT temperatures), but won't be massive enough to self-gravitate. Animatter is the least of the SMBb's worries, given all the hotter radiation surrounding the cold spot formed by condensing into the brain itself.
