I own a 77′ jazzmaster myself. It seems that there’s something wrong with the tremolo slot. When you insert the tremolo bar in the tromolo slot, the bar is simply locked by a metal cylinder which applies pressure on the bar. I can imagine that this cylinder maybe is damaged. The squeeking sound is probably the bar, hitting the metal inside the tremolo system.
The jazzmaster tremolo system is really easy. I think you should remove the system from the body yourself, and find out what the exact problem is. Just remove the strings and the system screws. After this, you can remove the complete system and find out what’s wrong.
Let us know, if your problem is solved!