Is there any justification for viewing neurons as self-interested agents?

The answer to "is there any justification" is pretty much always "yes", but that can be a quite uninteresting "yes" if those justifications are poor. People justify bad or unsupported ideas all the time.

A big difference between single cells of a whole organism and single-celled organisms is that only the single-celled organisms reproduce on their own. Natural selection helps optimize that process to preserve the cells (and organisms) which are best able to make new copies of themselves. In that sense, even multicellular organisms aren't really self-interested, they are "my next generation interested", which explains a lot of animal behavior that cannot be explained by self interest.

There are models of brain function that involve competition between neurons (like winner-take-all decision making, used both in artificial neural networks and to understand biological brains), but these algorithms aren't really "self-interest", they're just the way that a whole brain arrives at a conclusion. There is also competition in development of neurons and synapses (and in other cell types as well), but it's not really right to think of this as "self-interest", in particular because the "losers" undergo apoptosis - that's the opposite of self-interest, it's explicitly a behavior selected in cells that kills the cell for the benefit of the organism.

Could one still use a model of self-interest to understand these behaviors of cell survival and neural computation? Yes, probably, but it would be important to think of it as an analogy. The neurons in such a model might behave like a self-interested agent, but that doesn't mean they are one.

Sometimes cells in a multicellular organism do develop tendencies towards self-interest, when the genes that limit and control the cells' growth are not behaving properly. These cells start to do a great job of replicating themselves rather than functioning for the survival of the whole organism. We call this cancer, and it ultimately becomes self-defeating since once the whole organism dies, the tumor does as well.