Can someone back out of a purchase agreement?

The written agreement is the first and most important place to start. It's not possible to address your situation fully or fairly without reading it.
Generally speaking, let's say a buyer of real estate can demonstrate they have a proper contract, that they are ready, willing and able to close on the closing date and that they offered to to perform the contract. However, the seller backs out and refuses to sell. In that situation, the buyer might potentially be entitled to what is called "specific performance". This is a fancy legal concept that means the court may order the contract to be performed based on its terms.
If you are using an escrow company, then the transaction would not close without closing costs being paid (the escrow wants to be paid for their services).
Between where you are today and obtaining a court order for specific performance, assuming you would even win, is likely to be a long and expensive road. You're probably better off negotiating a compromise with your brother and making sure the transaction is done correctly (title search, title insurance, etc.).
Be well.

If he backs out of the deal you can sue him for specific performance and make him complete the sale. If the contract says he agrees to pay the closing costs you can remind him that you can most certainly enforce the contract. But, as my colleague points out, you may be better off trying to work with him to resolve it. There are a lot of potential issues that could arise, especially since you are already in the house. Remember, a lawsuit to force the sale will cost you quite a bit of money, probably a lot less than the closing costs. You can always ask for your attorney fees to be paid back since you are enforcing a contract, but that is ultimately up to the court. I