How deep can black boxes go under water?
They are required to be able to withstand a depth of 20,000 feet (3.8 miles, 6.1 km) for 30 days.
& how much pressure can they withstand?
At 20,000 feet deep the pressure is 8900 pounds per square inch (psi) which is 606 times atmospheric pressure at sea level (nearly 4½ tons on each square inch of surface)
A flight data recorder from AF447, around 13,000 feet deep for almost 2 years. Still readable
The DFDR must be watertight to a depth of 20,000 feet in sea water, and survive at this depth for 30 days
The units also have to withstand a static crushing force at all of its six axis points of an applied load force of 5,000lbs for 5 minutes on each axis.
See What happens to the transponder, FDR and CVR in case of an explosion?
The southern corridor of the Indian Ocean, where officials now believe the plane went down, ranges in depth from 3,770 feet to 23,000 feet. At its greatest depth, that’s almost equal to the height of Mount Everest (which measures 29,029 feet tall).
For every 33 feet (10.06 meters) you go down, the pressure increases by 14.5 psi (1 bar).