Yes, and in case anyone missed it, we have a thread discussing that here in P&E.
The Cenk Uygur case is much simpler. He is not a natural-born citizen so is clearly ineligible. There’s no ambiguity. The fact that he has still managed to qualify for the Democratic primary in some states just shows how some states have little to no say in who is allowed into a primary. I mean, in the case of the Democratic Party there isn’t even any ambiguity; barring something like a severe medical catastrophe or a sudden decision to not run, Biden is the nominee. The Democratic primary itself is a formality.
It really does vary from state to state. It has nothing to do with actual eligibility for the office itself. It’s a procedural matter codified (or not) into the laws of the states.