While FICA taxes (partially) pay for senior citizens' healthcare, your health insurance premiums both pay for your annual physical (especially on Platinum PPO plans) as well as care for sicker people that are not eligible for Medicare/Medicaid. I assure you that there are plenty of sick people that are under the age of 65, and actuaries price them in when computing premiums for private health insurance.
And as a minor point of order, even Medicare isn't "free" for seniors — the FICA taxes only cover Part A benefits, and seniors still need to pay monthly premiums to Medicare to cover Part B benefits. And even those benefits aren't 100% covered — Medicare only covers 80% of outpatient care.