i thought that if the individual electrons in a current had its own magnetic moment, shouldn't negatively charged particles be also possess that property ?

Yes, they do, and they will align in the presence of an external field. In order for them to create a net field, you would have to spin-polarize them so that all of the individual fields add up — if they are randomly aligned, the fields will cancel. This is the difference between ferromagnetic and non-ferromagnetic materials: you have unpaired electrons that can be aligned and because the atoms are in a lattice structure they can keep this alignment.