OK, I'm thingking two things from experience... I had the exact same proble; car wouldn't go into gear... BTW... any car will go into gear while it is off without even pressing the pedal.

It could be that your pedal, although pushed all the way down may not be fully actuating the plunger on your clutch master cylinder... this happened to me when I converted from auto to manual....

First....
There are 3 bolts that hold your clutch pedal in. Two come through the firewall from the master cylinder, you pedal slides over these two bolts then you slap the nuts on to hold the pedal against the firewall.

Now its the third bolt that I want you to look at to see if it is loose... this is where my problem was. The third bolt goes through the top of the pedal vertically into the underside of the dash area.

Mine was loose at this third bolt so when I pushed the pedal in the whole assembly would just twist and flex on the firewall. The third bolt holds it upright and stiffens the pedal assembly so that the pedal assembly does not move and flex with the firewall when pushed in. I tightened this bolt and voila... when I pushed the pedal in, it went right into gear.

Second, check you master cylinder and slave cylinder for leaks. Bleed the system. Check the fluids. You could have a leak or air in the lines which effects the hydrolic pressure. Even with a leak or air, you could still have pedal pressure but not enough to get the job done.

If you say that there is a squeeking sound that definitely sounds like your master cylinder. Mine is going bad because it is old and I'm running an ACT Heavy Duty Pressure plate... occasionaly when I push the pedal in, it will make a groaning squeek. Run your finger along the rubber seal where the master cylinders plunger rod hooks to your pedal inside the car... if you feel an oily residue than that too could be your problem; a bad master cylinder.