In my house, we use "non-school night" for Friday and Saturday evenings (Sunday evening is a school night, since it's followed by school on Monday morning). I think this would be very widely understood in the US, even for folks who are no longer in school. For example, the base term school night is defined by Cambridge Dictionaries (emphasis added) as:
the evening before a day when children have to go to school. Adults also use this expression informally to describe the evening before a day when they have to get up to go to work.
The negated term has even been codified into law in some places:
Non-school night always refers to Fridays and Saturdays but also means any night in which the next day Crisp County schools are closed to students.
—Cordele, GA Code of the City, "Curfew for Minors", Chapter 12, Article I., Sec. 12-13.
As these definitions suggest, while the term most often means Friday and Saturday evenings, it would stretch to encompass vacation nights in the middle of the week, but that would probably be clear from context. In context, it's probably less ambiguous than some more common phrases like "weekend evenings" with regards to the inclusion or exclusion of Friday and Sunday nights. It's also definitely a more casual, tongue-in-cheek expression when applied to adults, suitable for casual conversation but probably not for, say, company work policies.
So you could say:
I keep to myself on weeknights, but like to meet up with friends on non-school nights.
and be understood to mean that you like to meet friends in the evening and night hours of Fridays and Saturdays.
However, I'll note that I would usually expect a straightforward phrase, such as weekend nights or weekend evenings or in the evening on weekends or even Friday and Saturday evenings, possibly with some clarifying language where necessary. A bare on the weekend might even be sufficient in context; for example, absent any other information, I would assume a statement like "I like to party on the weekend" refers primarily to Friday and Saturday evenings/nights, since most "partying" happens in the evening/at night. That's coming from a US-cultural perspective, but I suspect the same now holds true in much of the Anglosphere.