In Ohio, can they stop your Medicaid because you smoke?

Medicaid in Ohio ( and elsewhere) is required to offer and pay for smoking cessation treatment.

Perhaps your neighbor was offered treatment and misunderstood?

No idea, why not ask Medicaid, but here's another thought too - if this notion does make your neighbour quit you'll be doing her a favour by playing along. Let discretion be the better part of truthfulness, perhaps?

And think of the money she'll save :)

Never.