Can my boss make me pay back training costs after I quit?
Agreements to repay your employer for training costs are valid and enforceable. So if you had agreed to repay your employer if you quit, you must do so.
Contracts are enforceable, even if one of the parties to the contract changes his or her mind about it, or even if it somehow seems unfair or excessive to hold a party to the contract--if someone agreed contractually to do something, they must do it. If they don't, they may be sued. That means that if there is a contract to repay your employer any training costs they paid for you in the event you quit, then if you do quit, you must repay them.
First, though, understand that if there is no contract to repay training costs, you don't have to. It doesn’t matter if you shamelessly took advantage of your employer and quit the moment you had completed some training you wanted and which they paid for. You do not need to repay training costs, regardless of the circumstances, if there was no agreement to do so.
But if there was an agreement that you'd repay if certain circumstances occurred, such as if you quit or resigned, then if that agreement formed an enforceable contract, you would have to repay. To form an enforceable contract, only two things are required: mutual agreement and consideration. "Mutual agreement" is exactly what it sounds like: you and your employer must both agree to the terms. For example, that they will pay for your training on condition that you will repay them the training cost if you quit. As long as both of you are "on the same page" in terms of what is expected or required, that's mutual agreement. That's why an agreement like this should be in writing. By having it in writing, it will be much more obvious what was agreed to, and there will less room for argument over what were, or were not, the terms of the agreement. And if the writing is signed by both parties, that will show that the employer and employee both agreed to what was in writing. (The law presumes that you understood and agreed to what you signed.)
"Consideration" is an exchange of things or promises of value. If the pay for your training, that is consideration from them to you--it is something of value. And if you agree that you will repay that money if you quit or resign, what you are really agreeing is that you won't quit or resign--that is, that you'll keep working for them as long as they want you. That implicit promise is a thing of value, and so is your consideration to them. Thus, a written agreement where the terms clearly are that you will repay training costs if you voluntarily leave (i.e., quit) will form an enforceable contract. If you do quit but don't repay, your employer could sue you for the money.
Agreements like this are enforced as per their terms--but only as per their terms. So say that the agreement is that you have to repay if you quit less than 12 months after the training (which is a fairly common type of agreement like this). In that case, if you quit 12 months and a week later, you don't have to repay. You only have to repay when the terms of the agreement say you do.