In addition to what the other posters have mentioned, if they transfer the house to you at less than fair market value -- tax consequences or not -- and they end up in a nursing home, that won't necessarily protect the house. If the government suspects that the reason they transferred the house, or disposed of any other assets, was to avoid having to use them to pay for nursing care, well, that's a fraudulent transfer, the government can undo the transfer, and folks can go to jail.
There are ways of transferring assets to protect them, but you'll need to enlist the services of an estate-planning specialist.