It's a big distortion of "Asylum seekers who are 65 or older and have a work permit based on seeking asylum, and have been working in the United States for 10 years or longer, may be eligible to apply for Medicare."

Immigrants and Medicaid & CHIP

If you're a "qualified non-citizen," you're generally eligible for coverage through Medicaid and the Children's Health Insurance Program (CHIP), if you meet state income and residency rules.
To get Medicaid and CHIP coverage, many qualified non-citizens (such as many Lawful Permanent Residents, also known as LPRs or green card holders) have a 5-year waiting period. This means they must wait 5 years after getting "qualified" immigration status before they can get Medicaid and CHIP coverage. There are exceptions. For example, refugees, asylees, or LPRs who used to be refugees or asylees don't have to wait 5 years. Qualified non-citizens who haven't yet had their status for 5 years generally qualify for Marketplace coverage, if otherwise eligible.
The term "qualified non-citizen" includes:

Lawful Permanent Residents (LPR/Green Card Holder)
Asylees
Refugees
Cuban/Haitian entrants
Paroled into the U.S. for at least one year
Conditional entrant granted before 1980
Battered non-citizens, spouses, children, or parents
Victims of trafficking and his or her spouse, child, sibling, or parent or individuals with a pending application for a victim of trafficking visa
Granted withholding of deportation
Member of a federally recognized Indian tribe or American Indian born in Canada
Citizens of the Marshall Islands, Micronesia, and Palau who are living in one of the U.S. states or territories (referred to as Compact of Free Association or COFA migrants)