In the absence of a contract, which I would be willing to bet you do not have, you can be fired for just about anything they'd like to fire you for.
Why in the sam hill would you refuse to TRY the test or refuse to do the extra training? Not only do you "have to" but if you refuse, and they fire you for refusing, you don't have much of a chance of being approved for unemployment or any kind of public assistance, because you just deliberately quit your job for a very poor reason. If they were asking you to do something dangerous or illegal, maybe that would be grounds to refuse to do what they ask of you. But just because it might hurt your feelings if you don't do well on the test, or you're hiding a secret inability to read or some other problem that would make it uncomfortable for you to take the test or you have "test anxiety" or something that's not a good reason to refuse to try.
If you do badly on the test I doubt if they'd fire you immediately. They legally could, they don't even need a bad test grade to fire you legally, they could fire you for showing up that day in green socks, but if you know you've done badly, or you are going to do badly, and you talk to them about it, they probably will give you other chances or accommodate your problems with the testing in some way. A good employee who tries to do what they ask is not someone they want to lose. It would be insubordinate to refuse a reasonable request from the employer that you do something, participate in some work related activity, or fail to cooperate with their normal training process. They can fire you with total impunity.
If you don't want to do what they require of you on this job, I suggest that you find another job where you are not going to be required to take tests or retrain in some way. That's your only right in this situation.