We present a Frank-Oseen elasticity theory for the shape and structure of deformable nematic droplets with homeotropic surface anchoring in the presence of a magnetic field. Inspired by recent experimental observations, we focus on the case where the magnetic susceptibility is negative, and find that small drops have a lens shape with a homogeneous director field for any magnetic-field strength, whereas larger drops are spherical and have a radial director field, at least if the magnetic field is weak. For strong magnetic fields the hedgehog configuration transforms into a split-core line defect that, depending on the anchoring strength, can be accompanied by an elongation of the tactoid itself. We present a three-dimensional phase diagram that shows the tactoid shape and director field for a given anchoring strength, tactoid size, and magnetic-field strength. Our findings rationalize the different shapes and structures that recently have been observed experimentally for nematic droplets found in dispersions of gibbsite platelets in two types of solvent.