I was wondering why some notations can be written with square and some can't.
There is no reason why a quantity cannot be defined in units of kg-squared.
For example, gravitational attraction depends on the product of 2 masses; this product could be given a name and quantified in units of $kg^2$. Likewise the magnetic force between two current-carrying wires depends on the product of the currents; this product would be measured in the unusual units of Amps-squared ($A^2$). This combination does in fact occur quite often within electo-magnetic units, but it has no special significance and no name.
The reason we don't do this already is that it is not useful, unlike $m^2$ for area and $m^3$ for volume, which are useful quantities.