Yes, your trustee can sell your home for you, but the proceeds of the sale must remain in the trust. If the trust document allows, those proceeds can be used to purchase a new home for you, but the new home will still be owned by the trust.
If the trust document allows those proceeds to be withdrawn, even if they are not withdrawn, those proceeds would not be protected from being seized by a nursing home or Medicare.
One of the main benefits of gifting your home to an irrevocable trust, not mentioned in the original answer, is this removes the value of your home from your estate. For most people, this is not vital for federal tax purposes, however, in New Jersey, if the value of your estate exceeds $675,000, you will pay estate tax to the state.