I don’t know where you got that info but the Portuguese government doesnt at all require anyone to have private health insurance.
You need a residency permit and to register in Portuguese social security. After that you take those two papers to a “centro de saúde” so that they can issue you a healthcare card. After that you have the same access to healthcare as any Portuguese person.
“Being a resident in Portugal, you are eligible for medical cards that provide free medical assistance from your doctor at your local health centre. In order to obtain the card, you are required to present your residency permit, along with your social security card to the health centre.” - https://www.allianzworldwidecare.com/en/support/view/national-healthcare-systems/healthcare-in-portugal/
Doctor appointments at the local health Centre (centro de saúde) are free and for anything else, exams, surgery, whatever it may be, may carry a small fee of about 20 euros
To be enrolled into social security in order to be able to request the healthcare card you need to go to a “Segurança Social” (social security) office and bring with you your residency permit as well as a document verifying your retired status
The US and Portugal have a deal since 1989 where your US social security status may be matched in the Portuguese system so that you can receive extra benefits and protections. You should probably contact the US embassy and the Portuguese SEF for details on how that works - U.S.-Portuguese Social Security Agreement | International Programs | SSA