Overheating and the squeaky noise will have likely tripped the fuse on your PI. This fuse apparently is slow to reset, so giving it plenty of time was the proper thing to do.
If your red light was flashing at all, that indicates a likely issue with the power supply. Double check that your power supply supplies 5 volts and at least 800 milliamps. If it still causes problems, replace it.
If your green light comes on but the PI still doesn't boot, then your SD card is somehow readable by the pi without anything for it to do. This is a fairly strange situation, but simply reformatting the card and reinstalling NOOBS should rectify the situation.
If you weren't doing anything intensive on the pi that might have caused the overheating, I suspect your power supply is faulty, and could damage your pi. If you were doing something that might have overheated the pi, get heatsinks, a vented case, or even fans to help cool it down. If you already are cooling your pi, then lighten up on the workload you're giving it! Its a tiny machine, and no one likes burnt pi(e)