First of all, let’s deal with your neutrals. It is against code and dangerous to have more than one neutral (“grounded conductor” in the language of the NEC) in a single hole/screw terminal of your ground/neutral bar. If it is the case that you have multiple neutrals in a single hole, that needs to be fixed.
408.41 Grounded Conductor Terminations. Each grounded conductor shall terminate within the panelboard in an individual terminal that is not also used for another conductor. Exception: Grounded conductors of circuits with parallel conductors shall be permitted to terminate in a single terminal if the terminal is identified for connection of more than one conductor.
Why is this dangerous? This post goes into great detail. (Re-quoting the important part here because the source link within is dead and can’t link directly:)
The connection of a neutral and equipment-grounding conductor in the same termination creates a similar issue. One of the objectives of the particular arrangement of bonding jumpers, neutrals and equipment grounds is to allow circuit isolation while keeping the equipment grounding conductor still connected to the grounding electrode (see UL 869A - Reference Standard for Service Equipment). When the neutral is disconnected, the objective is to still have the equipment ground solidly connected to the grounding electrode. If both the neutral and grounded conductor is under the same terminal, this cannot be accomplished.
Now, as to your Ground wires, it is fine to add an additional ground bar to the panel, by using the SquareD part for the raised attachment points which should be pre-tapped to make an appropriate ground connection, or by installing a ground bar using appropriate drill size, tapping, and making sure two thread turns of the screws mounting the ground bar engage with the (unpainted) metal of the box.
Note that you should use this extra ground bar for ground wires (“grounding conductors” in NEC parlance) not for neutrals. You can have multiple grounding conductors in the same hole, provided the equipment manufacturer says the hole and lug are rated for more than one conductor. See details in this post including some other good code references for why multiple conductors are permitted under a lug when labeled so. Here is the pertinent NEC reference:
110.14 Electrical Connections ... Terminals for more than one conductor .... shall be so identified.