"After I obtain my BSN, is there any doctorate or masters program where I would benefit from the experience of government nursing/correctional nursing?" Well...any experience is useful in rounding you out as a nurse...nothing specific to corrections although my PMHNP studies are highly relevant.
"Those of you with experience who have been doing this for a long time, where has correctional nursing or government nursing led you?" I have wonderful opportunities and love my job. My team provides medical support for CA inmates fighting wildfires. I have job offers regularly but that is in part because I maintain per diem work at an acute care hospital.
"What doors has it opened for you, compared to acute care, critical care, or bedside nursing?" I was the ADA at Kaiser's mental health in downtown LA for a while but went back to prison nursing because the retirement is better, less demanding, less stress, higher income....etc. Nice thing is I have 4 job offers as a mental health provider once my PMHNP and certs are complete, mostly through my state work.
"I know some graduate nursing programs require a certain number of med-surg experience, and I suppose that correctional nursing doesn't count in that regard." Don't sell correctional nursing short. I worked at a maximum security prison where I regularly responded to overdoses, stabbings, cardioverted a patient in our triage and treatment area (with a physician of course) IV insertion, IV medication administration, mental health crisis intervention, primary care, telemedicine...make the most of your chosen career and people will see it in your diversity of experience.