We study the capillary instability of infinite cylindrical jets of nematic liquid crystal polymers (LCPs) using an approximate Doi theory for rodlike LCPs with an emphasis on the role of surface elasticity, arising from the potential discrepancy between the averaged molecular orientation in the bulk and the preferred surface orientation. Our analyses focus on the linear stability of two equilibrium (constant), cylindrical jets of distinctive preferred surface orientation patterns: the equilibrium jet of the axially aligned uniaxial director corresponding to the preferred tangential (planar) aligning at the jet free surface and the one of the radially aligned uniaxial director corresponding to the preferred normal (homeotropic) aligning at the surface. For the axially aligning jet, our results establish that the capillary instability persists for a finite band of wave numbers and the surface elasticity lowers the upper bound of the unstable wave number band (cutoff wave number), reduces the growthrate across the entire band of unstable wave numbers, and lengthens the wave length of the most unstable mode compared to our previous results in the absence of the surface elasticity. We then extend the study to the radially aligning jet and show that it shares the same stability properties of the axially aligning jet qualitatively. Quantitatively, however, the radially aligning jet suffers from a much greater growthrate and shorter wave length at the most unstable mode with the same set of physical parameters. In both cases, we demonstrate explicitly that the cutoff wave number varies inversely with respect to the uniaxial order parameter of the underlying equilibrium jet. This study extends the 1-D asymptotic (slender body) analysis of Rey [18] to a full 3-D, axisymmetric analysis with variable degree of orientation and meanwhile our early work to include the surface elasticity [10].