I was recently ordered to move from one part-time position that I have held for 20 years, to another part-time position. This change includes continueing to do my original duties, in addition to work that was performed by another person that was fired. I was informed by management that I would lose my vacation and personal days, which I have been receiving since the start of my employment. Do I have any recourse under Ohio law; or if I quit, would I be eligible for Unemployment Compensation under a breech of contract clause?

The issue of vacation (or PTO) benefits are not addressed by Federal or Ohio state labor code. It's the company policy that matters.

You mention that you were employed under a contract. I'd have a lawyer review your contract to see if it's indeed "enforceable". If it's enforceable, your employer may be obligued to satisfy its conditions. If it's not enforceable, then your boss is free to change the terms; which I believe may be the case here.

Generally speaking you'd only qualify for unemployment if you are laid-off or terminated for no cause. Voluntary separartions rarely win UC benefits. It's not impossible; just extremely rare. IMHO I would not hold much hope in this situation.

It is very unclear that you have a contract and it is my best guess that you do not. Am I correct?

Do I have any recourse under Ohio law No. Your employer is free to change the terms of your employment as they deem appropriate; you, in turn, are free to quit if you no longer wish to work there.

or if I quit, would I be eligible for Unemployment Compensation under a breech of contract clause? What contract?