I was generally pleased with healthcare in Croatia while I was there (about three years).
I used both the public and private systems. For specialized care, such as gynecology, dermatology, and dentistry, I chose private clinics. For checkups and more general concerns, I used the public healthcare system. The advantage of scheduling private appointments is that you can often be seen right away, and the entire process is usually simpler than trying to find your way around a public facility. In my personal experience, there were more English-speaking employees at the private clinics than at the public facilities. As I do speak workable Croatian, this wasn't a huge issue for me, but there were a couple of confusing moments when I ended up at a hospital and didn't quite understand the routine.
I did not purchase private health insurance. Healthcare in Croatia – even private healthcare – is so affordable (at least compared to the US) that it wasn't a problem for me to pay out of pocket and then be reimbursed by my US insurance company later. I typically paid around 350 hrk for an appointment or an exam at a private clinic. Sometimes this included labs, sometimes labs were a bit extra. Public healthcare costs next to nothing. (If I remember correctly, there was a 10 hrk fee to see the family doctor. My emergency room visit at the hospital cost around 100 hrk, I think. Even with health insurance in the US, you can usually expect to pay at least $75 per ER visit.)
Croatian doctors did sometimes take health concerns a little less seriously than my doctor in the States might have, and they also were less apt to push pills – which was both a blessing and a curse. For example, I had a pretty persistent sinus infection, and I was told to get some saline nasal spray. Eventually it worked and my body healed itself, but there were moments when I just really wanted antibiotics. On the other hand, I found many Croatian doctors to be much more intuitive and sensitive to symptoms and holistic approaches than some of my American doctors. I had a particular issue that a Croatian doctor I saw once picked up on almost immediately, while my American doctors insisted that nothing was wrong multiple times. So I suppose the takeaway here is that experiences can really vary.
The best advice I have is to always ask for recommendations from fellow expats or Croatian friends who may be able to help you navigate the system.
Also, keep in mind that there are some excellent natural/herbal remedies available at pharmacies. A favorite was jaglac (primrose) syrup for a nasty cough and antiseptic cough drops for a sore throat. Pharmacists have great recommendations and can often fix you up until you can get in to see a doctor, if the meds they recommend don't do the trick on their own.