How do you cut stone with 45 degrees miter without having the stones break?

Use sufficiently hard and tough stones, like jade, which can withstand almost any type of sawing and polishing, even when very thin. Be sure you're working with high quality rough that is free of cracks. Use a high quality diamond blade saw, with a very thin and new blade. Be sure the vise and blade are perfectly aligned, and use plenty of clean water or oil lubricant. Saw the piece very gently! Pre-scoring may help, taping the edge may help. Another option for material that just won't cut nicely is to cut it oversize enough to allow for loss, and then grind it to size on a flat lap, again, very gently.