Halonium ions of alkenes can't be aromatic because, as you pointed out, there aren't any pi orbitals on the carbons.
As for the alkyne derived halonium ions, I think you can think of those as antiaromatic. The positive charge isn't residing in the aromatic system, it's in the plane of the three membered ring. Remember that to make the halonium ion the pi bond (2 electrons) add to the vacant orbital of the cationic halogen. The result is a three-center structure with only two electrons, the so-called 3-center, 2 electron bond that is common in organometallic chemistry. If you think in terms of electronegativity, most of the electron density will be close to the halogen and the carbon will bear most of the positive charge, and it clearly can't put the positive charge into the aromatic system because you already have a filled pi orbital there.