It is advisable to have insurance from an American company while in the United States, even if the premium for these plans are more expensive. The reason is that while almost all Doctors/hospitals in the United States accept American insurance company cards, they will be reluctant to acknowledge overseas insurance coverage. The medical office can easily contact an American insurance company for clarification, while the same will not be true for an overseas insurance company.
Typically medical offices in the US will bill directly to known American insurance companies. For overseas insurance companies you most probably will have to pay the bill, and then try to get the claim reimbursed from the insurance company.