A gluon string is a particular kind of open string terminated in two particles which are the sources for the field. Is it possible to have a similar arrangement with gluinos? At first glance, it seems to me that such object could not exist, or at least not as an extended object: you need bosons to mediate between the two sources, and you need bosons to build a classical field extended in space. On other hand, the fact that in the world-sheet such structure is just an object with fermionic coordinates --and without bosonic coordinates-- could be telling that you can build it, but it is not extended in space, just a point. But I have never read of such description, so surely intuition is failing here.
In SQCD, you can get "gluino hadrons", mesons and baryons in which the gluino is one of the constituent fermions. (Also a gluinoball, a glueball with some gluinos mixed in.) So you could have a "gluino string", but in the opposite sense to what you want: it's two gluinos, at the ends of a gluon string.
Another possibility would be a topological defect in a gluino condensate. Maybe you could get this in N=2 SQCD, with monopoles at the ends.
But what you're looking for is a "fermionic meson", two quarks connected by the "gluino string". Well, you can define this as an operator, a fermionic Wilson line. The only place I can find such an entity playing a role is in holographic approximations to QCD like the Sakai-Sugimoto model. In holographic QCD, you typically have color branes and flavor branes. A gluon is a string between color branes, a quark is a string between a color brane and a flavor brane, a meson is a string between flavor branes, and a baryon is a brane connected by strings to flavor branes (these strings are the valence quarks). These strings and branes all live in AdS space, and QCD lives on the boundary. Also, these are usually supersymmetric models - the exact, nonsupersymmetric holographic dual to QCD has yet to be determined.
In superstring theory, you do have fermionic strings, and they are extended in space; the Fermi statistics come from the fermionic fields on the worldsheet. And the meson strings in holographic QCD do have mesino superpartners. See section 3.3.2 here. They are discussing mesino operators in the worldvolume theory of a flavor brane. One is a quark-squark composite, but the other is a quark and an antiquark connected by a gluino (the adjoint fermion). So we had to go into anti de Sitter space to find it, but there it is, the fabled fermionic gluino string. :-)