Since father was American and mother wasn't, there would have to be a DNA test performed to confirm Donald as a father, but with that, Barron could get American citizenship. He could now no longer get Slovenian citizenship though, as although Slovenia does allow dual citizenship, both must be in European Union. Unsure how those who already have US and Slovenian citizenship are able to continue having them... I guess they are grandfathered or nobody cares about enforcement, or possibly Slovenian government just isn't informed of their other citizenship.
Not like he would want it. Just saying this as his mother is Slovenian and this post is about citizenship...