The vast diversity in neuron cell morphology has led to an increase in automated algorithms which can accurately reconstruct neurons from microscopy images. The poor quality of brightfield and fluorescence microscopy images and the thin branch-like fibrous structure of neurons make the process of manual segmentation challenging. We propose a novel automatic neuron segmentation framework using a B-spline based active contour deformation model with hyperelastic regularization, and develop a MATLAB software tool named "NeuronSeg_BACH". In NeuronSeg_BACH, initialization of the contour is done automatically by detecting cell body and neurites separately. This boundary-extraction based algorithm utilizes cubic B-splines to deform active contours to match the neuron cell surface accurately. Using adaptive local refinement, finer level deformation of the active contour is captured using truncated hierarchical B-splines in a multiresolution manner. By introducing hyperelastic regularization, we allow large nonlinear deformations of the active contours. Unlike other existing methods which represent boundaries as piecewise constant functions, we provide a more accurate and smooth representation of the neuron geometry. In the level set segmentation framework, the implicit level set function is defined using $C^2$ continuous B-splines. Improved segmentation results are shown for 2D and 3D synthetic and microscopy images as compared to other methods.