I'll have to agree that the engine might be tight. Judging from how quickly it shuts off, no wind down period, just an abrupt stopping. You could be looking at a bearing problem or just a very tight fit of the rings. Heat would exacerbate the problem. As things warm the clearances get tighter.
I have seen new engines come from the factory with a piston ring not installed correctly and was bound up in the cylinder. I've also seen defective or improperly installed bearings. The squeaking sound you hear might be an indication of either of these conditions.
Remove the spark plug and the left side (magneto) cover, disengage the clutch and see if you can rotate the crankshaft by turning the magneto rotor (magnet). You should be able to do that with little effort. If not, you'll need to start looking deeper or consider returning the engine to your supplier. Some warrantys, if you have one, might be void if you disassemble the engine. But that is another can of worms.
Let us know what you find.
Tom