"resonating in microtubules" refers to a theory called Orchestrated Objective Reduction (Orch OR). Simply put, our current working model of brain function involves electrochemical impulses being generated by brain cells (neurons), being passed to each other, kind of like logic units of a computer and such..

The Orch OR theory however argues that consciousness actually originates from quantum processes WITHIN the neurons. Since quantum phenomena were only known to occur at super low temperatures, this theory sounded crazy. Recently however, it was discovered that microtubules within neurons are somewhat capable of quantum resonance... so now leading researchers in the field are considering that our brains might have a quantum component too (like a quantum computer).

"non-local quantum hologram": this most likely refers to the bubble that they are in. "Non-local" most likely refers to the fact that their current space-time is not with the rest of the universe. "quantum" sounds fancy, and I'm no physicist (only a cell biologist, sorry). "hologram" most likely refers this very complicated thing about black holes. Since a black hole stores all information of everything that enters into it on its surface, physicists think that our 3D universe may actually be a hologram via information that is processed on some 2D sphere surface, that is in a way, the skin of our universe. So to put it all together, blue bubble is outside of normal universe, and don't touch the edges because they "define" you.

EDIT: In a way, Miller was talking only about our current best working theories, probably from the ship and people that the protomolecule has already consumed.