I was driving yesterday, and the car just died on me - like it ran out of gas. I pushed it into a parking space and tried several times to start it but it would not crank. It would turn over like it was about to start, but wouldn't run. I had it towed to a mechanic, and they said they started it right up this morning and it's been running fine. The only code they could get was for "multiple misfires." The only unusual thing I could mention is that there was a high pitched whining noise and a sound like a squirrel cage motor running under the hood on the back passenger side of the engine compartment. I did have the oil changed a couple of days ago, but other than that, nothing unusual.

Any ideas what could be the cause of this problem?

Could be your alternator going bad. All the symptoms you mentioned fit. Helped a friend out by replacing her alternator because her car died like it ran out of gas and then would not start up again right away but did after it cooled down. Sometimes it was even having trouble firing up when it was cool. It also had a whining noise going on, which turned out to be the alternator bearing, and it had multiple misfires. Unfortunately, she waited too long to get it fixed and only asked for my help after it started "backfiring" as she stated it. The "backfires" were actually accumulated fuel in her catalytic converter getting ignited when the cat got hot enough. This was unspent fuel pushed into the exhaust system because the alternator intermittently failed to generate power to fire random cylinders at the combustion stroke. The engine, still running but badly, then exhaust all that fuel that didn't fired off. I replaced the alternator and it ran great after that, except for the catalytic converter which was now wrecked because of the internal combustion within the cat tore up its insides. The alternator was cheap compared to the cost of a new cat and it could have been saved early on.

Incorrect fuel pressure and intake system vacuum leak can also cause misfires.

On another VQ engined car I experienced a camshaft sensor if going bad will make the car just stop running.