1998 Toyota Tercel the engine of my car stopped, not sure w

1998 Toyota Tercel Automatic 220000 km

Last night I stopped at a red light and the the engine just stopped working then and there, the engine check and the battery lights came on. When I tried to start it again the motor wanted to start, there was just a squeaking sound and wouldn't start. My friend was almost sure it was the alternator. However I towed it to a garage and they told me, that some oil switch was broken and the oil spilled out. They said there was no oil in the car, and when they put 3.5 litters in it, it started, so they changed that oil switch thingy. My question is if I had no oil in the car wouldn't my engine totally block? Wouldn't that actually burn the engine?
I just want to mention that I did not notice any oil trace on the ground, neither on the motor. I'm worried that the mechanics didn't fix the actual problem and my car will give up on me again.