How should I implement Default Associated Values with Swift Enums?

I know this is a bit old, but would this work for what you want?

typealias FilterIdentifier = String

enum DefaultAPIFilters: FilterIdentifier {
    case Everyone = "everyone"
    case Team = "team"
}

enum APIFilters {
    case Default(DefaultAPIFilters)
    case Custom(FilterIdentifier)
}

let everyoneFilter = APIFilters.Default(.Everyone)
let teamFilter = APIFilters.Default(.Team)
let clownFilter = APIFilters.Custom("clowns_only")

Extending Nathan Perry's response:

You can add a

var underlyingString: String {
  return getUnderlyingString(self) 
}

to the enum. Then define

func getUnderlyingString(apiFilter: APIFilters) -> String { 
    switch apiFilter {
    case .Default(let defaultAPIFilter):
        return defaultAPIFilter.rawValue
    case .Custom(let custom):
        return custom
    }
}