Can a public university block me on their official social media account if I am a student and/or employee?
Can a public university block me on their official social media
account if I am a student and/or employee?

De facto: yes. De jure: we don't know (see Noah Snyder's answer and the comments), but I don't see a compelling reason against it.

I am concerned about potentially starting a legal battle with the
university

Then don't.

But I'm not sure I should even worry about it because of how close I
am to graduating and whether it will affect my job search.

Starting legal battles simply because of political disagreement or because you cannot accept being blocked (after a twitter fight with you using multiple accounts) means there will be a public record of behavior which many people will find unpleasant. Such a reputation can obviously affect future job prospects. To avoid this, don't do anything which earns you a this reputation (i.e., let it go).

At the same time, even if it will have no long-lasting effect on me, I
don't want the problem to go unaddressed for other students.

As you only found out &quot;rather accidentally&quot; about being blocked, there seems to be not that big of a problem. Possibly other students avoid this situation by avoiding posts which get them blocked in the first place.