I don’t know about Texas in particular (for that you’ll have to check with a Texas elder law attorney), but where I practice (in Massachusetts) if you were to go to a nursing home and apply for Medicaid, you would be presumed to be the owner of the account and have to spend the funds on your care before Medicaid would kick in. This is a presumption and can be disproved with evidence that the account belongs to your son, but the burden of proof would be on you. Unless you find that the rule is different in Texas, it may be safer for your son to remove your name from the account and to appoint you as an agent under his durable power of attorney so you can step in if ever necessary.
For an article on the dangers of joint accounts, click here.
For more on Medicaid's asset rules, click here.