When installing a new load center should I add a ground bar dedicated to the grounding rod. The old box had the ground tied to the neutral in the load center and the new Square D panel does not have a dedicated ground strip.

If this is the main panel then no. The grounds and neutrals share the same bar(s) in the main panel.
Make sure the (green) bond screw is installed.

INMO, there are two reasons to have a separate ground bus and I think the Sticky about grounding has some information in it.

1. Asthetics. Te wiring might look neater.
2. In critical applications there are 2 grounds. Green and green with yellow stripe.

Signal and safety grounds: single point ground

That's effectivey what your doing when you add a ground bar to a main panel. You isolate, somewhat, the safety and protective grounds and you tidy things up a bit.

Hospitals will have orange receptacles which have an isolated ground. Outlets are typically wired with conduit.
To get an isolated ground, the outlet would be home runned to the panel in a typical house.

Hospitals will keep signal and protective grounds on separate busses.

This isn't required in residential and hardly no one does it.

Have you contacted the local power company to make arrangement for a higher amperage, guessing that that is why you want to up grade the panel

Also, you need to make sure you have the appropriate permit from your LAHJ.

If you do not understand the concept of "bonded" then please contact a licensed electrician to help you. Replacing main service panels can be deadly if you do not know what you are doing.

What is the exact question you are asking here?

Are you asking whether the main service panel requires the presence of a ground rod. The answer is "Yes."

Do you need a EGC (Equipment Grounding Conductor) to connect the Main Service Panel to the ground rod, also, "Yes."

If you are asking whether there needs to be a Neutral Bus Bar and Ground Bus Bar the answer is in the design of the Service Panel.

It mandated by the NEC that at the Main Service Panel (ONLY) the ground and Neutral buses are "Bonded" together.

Therefore, if the manufacturer does not include two separate bus bars one for grounds and one for neutrals then the bus bar that is provide can be used for both.

"Some" prefer the use of separate ground and neutral bus bars to provide a more aesthetic appearance to the wiring in the panel box. Not my choice but others like it.

The grounding system provided by the EGS to a Grounding Rod connection is for a completley different reason but it is required.

For the life of me I don't know why some folks think this is more aesthetic, and I would certainly not say "most".

I actually hate when the ground and neutrals are intentionally kept separate in a main panel because this creates a situation where whites and grounds are crossing over from side to side which really looks messy.

Don, you are certainly OK. I was just expressing an opinion based on my observations.