Oral contracts can be enforceable although the existence and terms of an oral contract may be hard to prove. Each state, with the possible exception of Louisiana with its civil law tradition, has the equivalent of an ancient British law call the Statute of Frauds, which requires certain contracts to be in writing in order to enforceable. Although these vary from state to state, typically real estate contracts, contracts to be performed in more than a year or contracts for the sale of goods costing more than a certain amount are required to be in writing. You should talk to a South Caroline attorney to learn what the requirements are in your state. I would point out that just because you don't have a formal signed contract doesn't mean that there are things in writing that support the existence of a contract. Letters or email messages that discuss the arrangement, receipts or deposit slips for payments made, and such could be evidence of the existence of the contract. I might also inquire as to the immigration status; it is possible that the brother had to sponsor his sister for a visa, which may involve certain obligations and commitments. I recommend that you or the sister collect all of the documentation that might relate to the arrangement and then see a local attorney.
Family - sometimes you can't live with them despite the best intentions.
We need to take a look at the cause of the situation since ther is no clear way to enforce this verbal contract. Why bring her over from Greece and incur all of these expenses. Did she get paid wages for her work? Did she file a tax return? Bottom line - did she benefit at all from the arrangement? If not, you would have a claim, but then you need an attorney.
If she did benefit look at what caused the fight? It would be easier to fix this problem than to spend a lot of money litigating something that would leave both parties out more money.
Work this out. It's the better choice.