Well, all of this is really kind of a moot issue, IMHO. Government’s, especially one with the material resources stockpiled like the US Federal Government, really don’t just evaporate overnight, they just get meaner, harsher, and more desperate.
History gives a hint at what would most likely happen.
If things got bad, most likely the prisons would go on lockdown, and the inmate population would be used as a slave labor force. At some point they would serve as a recruiting ground for live bodies to fill uniforms.
If a prison got overrun by, say, a rebel army, then they would either be shot in their cells, liberated and set free, recruited to fill the ranks, or any combination thereof. The approach of an enemy army is generally the only time guards abandon prisons, and then it’s a toss up on what they will do.