The NDA I will have to sign contains a part about disallowing any public disclosure of personal opinion about the company and some company related things during employment and indefinitely after termination (paraphrasing). I don't have much experience with this so I wanted to ask if this is something that is done routinely or if this is unusual since this seems a bit crazy to me.
This is pretty normal, at least in my experience in US companies.
Often times the company needs to protect its intellectual property and/or shareholder value by disallowing public disclosure of unpublished information, disparaging comments or even employee titles and contact information. Quite literally, Confidentiality and Non-Disparagement clauses.
Not only will you see such clauses in your typical NDA, but in the event of termination or release these will be re-enforced, potentially as part of a separate severance or release agreement (which also typically makes it so you cant poach employees, sue for wrongful termination, or sometimes contains a non-compete clause).
As an aside, it's important for employees in such companies looking to protect their IP to contain themselves anyways. US IP law works something like this: if you accidentally make a claim public knowledge, it makes whatever it is harder to patent, because the claim is no longer proprietary. All the company can do is sue you, and thus such slips can cause large monetary damage. Disparaging comments can also lead to PR nightmares that lose shareholder value or investors. So these contracts will look quite severe.
As always, though, have a lawyer review contracts you're uncomfortable with.