The alt isnt squealing, the belt on the alt is. Many people dont realize that an alt shows more and more resistance as its output goes up. Spin an alt when its out of the car and it spins freely, only resistance being from the internal bearings. But once its hooked to the engine and outputting power, the alt itself creates resistance.
Many people understand how an electrical motor works, but fail to apply that knowledge to an alternator, which is basically a motor in reverse. Elec motors work by alternating current thru them to reverse the polarity in the windings. The stator is a stationary piece that the magnetically charged windings react to (either pulled towards it, or repulsed from it), which creates the rotational force outputted to the motor shaft. An alternator works very similarly, but in reverse. The windings produce a reversible magnetic charge, just like a motor, but electrical energy is created when the force of the engine spins the assembly and creates the magnetic field. So, the more amperage the alt must produce, the more resistance it will provide against the spinning force from the engine.
Knowing this, your stereo has forced your alt to start producing more power than it previously had to before you installed the stereo. Again more power from the alt means more resistance to it spinning. This resistance is exerted to the engine via the drive belt. If the drive belt is squealing, it means the belt is slipping on one of the pulleys (creating the squealing noise). In other words, your problem may be as simple as tightening your belt(s). If you do this, and you notice your system voltage is still getting lower and lower, then its time to look at other options such as a bigger alt. Carry jumper cables with you.
Doing the 'big 3' means making your charging system transfer than electrical energy more efficiently throughout the car. This can obviously help, or hurt, an insufficient current supply situation. I almost always recommend upgrading the big 3 for this reason.