Until my family members and I individually ended up switching to insurance with nice low flat rates for services, I used to negotiate everything from office visits to ER bills. Before service if you can (especially for elective/outpatient type things), otherwise they really like cash in full. They tend to prefer working outside of insurance too, since it means more money for them even if they're halving your bill.
I'm sure there are services, and occasionally you'll find more tips on places like reddit (/r/frugal comes to mind). I have no immediate suggestions for how to do this in a time of crisis - it mostly depends on someone that knows you well giving enough of a shit, and when they'll try to bill you for services. :(
Why this is a thing is beyond me, but while it's the case...
IMHO negotiation is one of the problems causing high health care costs. Prices are crazy high by default, but insurance companies are able to negotiate lower prices from providers. This means the uninsured are billed higher by default. It turns out that individuals can negotiate lower prices too - sometimes. I advocate that a given service provider should be forced to charge the same price for the same service to all patients. This might result in competition and lower prices, but in the mean time it should lower costs for the uninsured.
I know of these largely through news coverage, and cannot speak to how effective they are (or what the costs are).
The stories I've heard, though, of the absolute complete and total nightmare that is the US healthcare scam, are heartbreaking.
I'm looking for online references but cannot find any at the moment, maybe others here have more information.
How does one pay attention to this in a time of crisis? Is there a good guide to navigating it somewhere?
Last summer I woke up with a monster headache and suddenly stopped being able to put words to thoughts. Tried calling a taxi, couldn't say where I wanted to go, so he called 911. On the way to the hospital in the ambulance I was just praying I was making the right decisions. The paramedics couldn't understand me and that frustrated them. They asked me what hospital, being an intern and totally new to the city I wouldn't know what to tell them even if I could. I was worried they'd treat me like I was on some kind of drug.
Turns out I had contracted a meningitis. The ambulance ride was $1000.