We study flux tubes on Higgs branches with curved geometry in supersymmetric gauge theories. As a first example we consider N= 1 QED with one flavor of charges and with Higgs branch curved by adding a FayetIliopoulos (FI) term. We show that in a generic vacuum on the Higgs branch flux tubes exist but become "thick". Their internal structure in the plane orthogonal to the string is determined by "BPS core" formed by heavy fields and long range "tail " associated with light fields living on the Higgs branch. The string tension is given by the tension of "BPS core" plus contribution coming from the "tail". Next we consider N= 2 QCD with gauge group SU(2) and N f = 2 flavors of fundamental matter (quarks) with the same mass. We perturb this theory by the mass term for the adjoint field which to the leading order in perturbation parameter do not break N= 2 supersymmetry and reduces to FI term. The Higgs branch has Eguchi-Hanson geometry. We work out string solution in the generic vacuum on the Higgs branch and calculate its string tension. We also discuss if these strings can turn into semilocal strings, the possibility related to the confinement/deconfinement phase transition.