Your foot pressure on the brake pedal, say 50 pounds for discussion, gets multiplied about four squillion times by leverage and this energy is exerted directly upon the brake fluid inside the master cylinder.

When you move the MC piston, the o-rings on it slide past the opening from the plastic fluid reservoir that keeps the MC full of fluid. It keeps the refill tank out of the high pressure zone.

With respect to squishy, drive the car a few days with the squishy pedal, beat the brakes up some and then bleed again. If it improves but you don't have a rock hard pedal, repeat enough times so you are absolutely certain you have all the air out before considering parts replacement.

There are about three thousand threads that say the exact same thing I have said in the above paragraph. It is not you doing a crap bleeding job or having failing equipment, it's the nature of the beast.

Shake it up and try again. Keep us posted.