100 euros ish for the year, just for immigration purposes. (But it's real insurance, covers you for emergency treatment, or getting you back to USA. It means Bulgaria won't have to pay if you get sick, that's what they want to know.)
I don't think USA medicare covers international, but you should check this.
You can get private health insurance in Bulgaria, just as you do in USA. But it's a LOT cheaper!
You can also pay into the Bulgarian social security system (I registered as unemployed, it's about 13 euros per month). Once you've made a few payments you're fully covered and entitled to free healthcare in Bulgaria.
This state cover works even in private hospitals, so they get something from the government, and they charge you the extra (not much usually) for the private room / VIP treatment. I've usually just paid the extra myself, but your private health insurance will cover this for you (and the policy will be cheaper than if you're not in the state system).
I think public hospitals are fine, but the private ones just look a bit fancier, and they have less people waiting.
I have a very nice private hospital (Pulmed) across the road from my apartment in Plovdiv. I'm so lazy, I don't even have a local doctor... I just walk across and get a consult at their emergency room. Last time, I had a 5 minutes wait... got an English-speaking doctor. Emergency room doctor consult + EKG + ultrasound + blood pressure + blood panel... total cost (fully private, no state cover, paid in cash by me) = 75 euros. I doubt you can even get a blood panel in USA at a hospital for 75 euros, let alone get fast turnaround with results in 30 minutes or whatever my Doc asked for.
I also went and did research recently on knee and hip replacements at the top private hospitals in Sofia. They said they will talk directly with state system (I don't have to do anything) and they will get the state standard contribution (as I'm now fully covered). No waiting list (next week probably, if I wanted). Cost for top-of-the-line athlete's hip implant (titanium stem, dual mobility ball, XPE, the biz), private room, VIP treatment, couple of weeks rehab included... extra 2k euros. Have a look what that would cost in USA!