I guess that depends on the level you’re talking at. They both interact with each other, so in that sense yes, but otherwise the interactions are quite different.
For the electrons in the double-slit experiment the interaction is between the electron wavefunctions exiting to spatially separated slits, making an interference pattern apear. This is a very simple process, albeit with far reaching consequences, and can be explained nicely with wave-mechanics.
The self-interaction of the Higgs, or of any other particle, is far more fundamental, as its an intrinsic property of the field (you cannot change the interaction, say, like you could by separating the slits in the Young’s experiment). For example, a similar self-interaction takes place with electrons, and changes both their mass and electric charge (with the help of photons, quanta of light), giving us the electrons we all know and love.