Not so different than in the States, e.g., 1) Will all US health care accept all forms of health insurance? 2) Will all US health insurers pay for all sorts of care at all sorts of US healh care providers? Nope x 2.
Which type of care? True emergency versus regular illness? Regarding your 2 very broad questions, the answers depends on a LOT of things x 2. Reasonable responses include maybe, maybe not, sometimes, & nope.
Depending on the form of care, sometimes the cost is low enough you just pay. On the other hand, some foreign hospitals require payment up front. Then, a US-insured patient fends for themselves to potentially be reimbursed by a US insurer.
1. If you've not been, go to Ireland. Amazing country. Love it there.
2. My one experience with the health care system there was very positive. Wife came down with full blown pneumonia on our first day of a trip. We were staying in a B&B, and when she came down to breakfast, our hosts could tell she was obviously very sick. She called her MD, and drove over while we followed (we were checking out). He saw her right away, diagnosed and prescribed antibiotics (z-pack). Cost was about $60. We went to pharmacy and got the prescription. Cost was about $20. No insurance involved.
Dialysis is clearly more significant, but those cousins should be able to provide details on what you'd need to do.
Typically, you will need to pay out-of-pocket up-front for any medical issue while traveling abroad and you would then need to seek reimbursement from your primary US health insurance provider upon your return, which they typically reimburse 60%. Heaven forbid there its a major issue and you need medical transportation back to the US, it can get really expensive. I am going to Ireland soon with my wife and purchased individual primary medical coverage for the two of us with a deductible of $250 for 12 days of coverage. The cost was just under $80 through UnitedHealthcare. Aetna, GeoBlue (Blue Cross), and IMG are other insures that provide similar coverage.