The Temperature of the core is nearly 10,000°F. This is far above the melting point of Carbon, so, no.
Also, diamond is a lot less dense than magma, so it would float rather than sink.
Also, if a human being fell into a lake of lava, they wouldn’t sink either, since humans are less dense than lava. You’d float well above the surface, at least until you get turned into mostly CO2 and H2O. Lava is a lot denser than water, so objects in lava don’t react the way we expect.
Also note that there have been flood basalts in the past where vast areas are covered with lava that just poured out of the earth. The Siberian Traps covered over a million square kilometers. However, all that lava didn’t pour out at once, it probably took hundreds of thousands of years to cover that area. Nobody’s really sure what causes these, aliens with super-magical bulldozers is a new hypothesis.
Barring things like impacting continental plates, most of that “squeeze” below the surface of the earth just comes from all the weight of the stuff directly above it. So your juice box analogy is fine, but there’s no external squeeze. The only thing providing pressure is the weight of the juice. If you insert a straw in the juice box , even though there’s pressure at the bottom, its only enough to push juice in the straw back up to the level of the rest of the juice (Funny how that works).
Depends on the pressure, too, don’t forget. It’s possible for planets to have diamond cores with temperatures far in excess of 10k °F. Just requires a considerable amount of pressure.
The Daleks tried it in the 22nd century, didn’t work out too well for them.
Sure, but assuming my penis is not available, how else would you accomplish this? 
WARNING! Do not attempt to drain the center of the earth!
The drilling machine you use would be destroyed, as would the planet itself. Any data collected from the experiment would be lost. And who would be crazy enough to fund you?