I was a member of a private club but can no longer afford to pay the dues to renew my membership. Can this club prevent me from coming to events that are open to the public?
Probably. A private club can usually decide who it wants to admit to its programs without being subject to litigation about the question. A “private club” is by definition not a “public accommodation” that might be subject to non-discrimination laws, but even if it were, I am not aware of any public accommodations law that includes “former members” as a protected class.
I don’t know whether you have explained your situation to the leaders of the club or whether you plan to renew your dues when you have the ability to do so. They may think you are trying to free load off them and might have a different view if they know what is going on. Or they may say you can’t pick and choose what events you will participate in if you haven’t paid your full dues. As a private club it can pretty much set its own rules.