If there are only negatively charged particles, then they cannot really form an atom, the repulsive interaction between the particles will push the particles away from each other, hence they cannot form a stable atom.

As a principle- systems like to maintain a neutral charge, this is why in normal atoms the positive protons attract a cloud of negative electrons. In your proposed atom of just negative particles, they would all repel each other, meaning an atom would no longer exist. This is a rather non-physical situation when viewed as an atom, but it is very similar to the flow of electricity. When something causes too many electrons (i.e. negative charge) to build up in one area, the electrons flow away form the high negative charge towards a more positive area, which makes electric current.

Unless the negatively-charged particles are antiprotons, then it wouldn't be an atom. Atomic nuclei require the strong nuclear force to hold them together, and apart from antiprotons, no nucleons have negative charge. Antiprotons are antimatter, so an atom consisting of a single antiproton is simply an ion of anti-hydrogen, and behaves exactly the same way as normal hydrogen, except that it's antimatter instead of matter. To make a bigger antimatter atom, you would need to add antineutrons as well as antiprotons, but antineutrons, like regular neutrons, have no electric charge.

Electrons cannot form atoms by themselves. To have an atom with electrons, there must be a nucleus that creates electron orbitals, and that nucleus needs to be held together by the strong nuclear force, not the electromagnetic force. Electrons happen to be negatively-charged, but that's not why they cannot by themselves make an atom.

If there were only negatively charged particles in the atom, the particles would repel one another because like charges repel each other (so positively charged particles would repel each other, too), and the atom would break apart. Part of the reason that atoms as we know them stay together is that the positive and negative particles are "glued" to each other because of the electromagnetic attraction between positively and negatively charged particles.

The negatively charged particles would never get together, because they repel each other.