Re: Non-spousal joint tenancy of stock account?
Nords and Martha make a good point with the TOD or POD, if the institution accepts these designations.
Joint tenancy creates all sorts of problems -- creditors, divorce, loss of control and ownership.
Re: transfer to qualify for medicare. Medicare does not cover the nursing home costs, except for a limited period of skilled nursing home care in specific circumstances. Medicaid does, but there are a number of issues revolving around legality, morality of the divestiture on the one side, as well as the fact that nursing home care on the public dollar may not be the quality of care that is available on the private dollar.
My suggestion is to contact an estate planning attorney who specializes in elder law. The first meeting, to get answers to your specific questions, should not cost more than a few hundred dollars, and may well be gratis.