  Under the presence of external magnetic fields with cylindrical symmetry,
Skyrmion-string defects with arbitrary topological charges are shown to appear
in spinor $F = 1$ Bose-Einstein condensates. We show that, depending on the
magnetic field boundary condition, the topological spin texture, at the planes
perpendicular to the cylindrical axis, can take zero, half integer, or
arbitrary values between $-1/2$ and $1/2$. We argue that these are true
topological defects since their charge is independent of the spatial location
of the singularity and since the total Skyrmion charge is the sum of the
individual charges of the defects present. Our findings are obtained by
numerically solving the corresponding fully coupled Gross-Pitaevskii equations
without any symmetry assumptions. We analyze, both, polar $^{23}$Na and
ferromagnetic $^{87}$Rb condensates.
