How would that help? Whether it's written or not, the agreement is as you say, until one party decides to end it.
And even with such an agreement, they might still ask you to sign something on day one. If you don't sign it, they may very well decided to "terminate the agreement" on day one.
If you're fired, you're eligible for unemployment. You're also eligible if you quit your job for a new company but then they rescind the offer. That's basically as much protection as you get in the US, since most states are at-will.
