The key to whether owning your house will affect your Medicaid or SSI eligibility is whether you live in the house. Both Medicaid medical coverage and Supplemental Security Insurance -- SSI eligibility depend on having very low income and assets. But if you live in a house you own, the house will NOT count as an asset when Medicaid and SSI decide on your eligibility. So, owning the house will not affect either your eligibility for Medicaid or SSI, or the amount of your benefits, as long as you live in it.
Also, neither Medicare nor SSI ever have a claim on your house or any other property, whether you live in it or not. Medicaid is different, but still protects you as long as you're alive. Medicaid might place a lien on the property to recover the cost of long-term care you might receive in a nursing home, if you ever enter one and Medicaid pays for it. But this only applies to long-term care, not to regular Medicaid medical coverage. And Medicaid cannot force the sale of the house. It merely places a claim -- in the form of a lien -- against the house to collect from its value after you die for the amounts it has spent on your nursing home care.
If you ever sell the house, you have to be careful that any money you receive from the sale does not disqualify you from Medicaid and SSI. If you sell the house and keep any of the money from the sale, that money would be considered by SSI and Medicaid as part of your assets, and could disqualify you as long as your total assets are above Medicaid and SSI eligibility levels. If you plan on selling the house and buying a different house to live in, you have to be careful not to have any delay between ownership and occupancy of the two houses. If there's a period of time when you have cash from the sale of one house before buying another one, you might become ineligible for Medicaid and SSI until you use the money to buy then next house and move into it. Also, if the house you buy costs less than the amount you receive for your former house, the amount of money you keep becomes an asset and could affect your eligibility.