Generally speaking everyone who is working in Hungary can be insured (employed, full-time self employed, farmer, church personel..) no matter of the status of the individual.
As a general rule, insured persons are entitled to all social security benefits, including cash benefits in addition to benefits in kind, thus providing an opportunity to mitigate loss of earnings in the event of sickness and maternity. Insurance is valid from the beginning of the underlying legal relationship until its termination (losing the job)
It is important that you loose your insurance right after the day when you stop working unless you are considered a “resident for social security purposes” (being a refugee, beneficiary of subsidiary protection, Hungarian citizen, or permanent resident), when you have a 45-day waiting period.
For a longer list of who can get a TAJ Card by being insured: https://www.neak.gov.hu/felso_menu/lakossagnak/ellatas_magyarorszagon/jogosultsag_az_ellatasra/az_egsszsegbiztositas_szolgaltatasaira_jogosultak/biztositottak_penzbeli_es_egeszsegugyi_ellatasra