Hi - first post, so mods please let me know if this isn't cool...
In a word, no, you cannot grow into specific shape.
There are two completely different ways to grow large single crystal diamonds - High Pressure, High Temperature (HPHT) and Chemical Vapor Deposition (CVD). HPHT results in cuboctahedral crystals, and there is a fair amount of variance in the finished crystal shape, and a fairly high failure rate. CVD diamond is grown on top of a plate of diamond, and the resulting diamond is in the basic shape of the starting plate (think snow on a patio table).
Check out the season 2 finale of Explained on Netflix - I discuss the growth process in detail and they put together some great graphics.
That being said, you can use a laser 5 axis CNC machine to cut more or less any 3D shape you want out of a lab diamond.
Now to 'well actually' myself, you can adjust the parameters of diamond growth and/or the dopants added to the carbon to get some variance in the shape of the resulting crystals. You can also change the number of seeds you put in the growth chamber depending on what general size you want to grow.