Your current code returns a Stream<String>, so you need an extra step to return a string:
Optional<String> auth = cookies.stream()
.filter(c -> c.getName().equals("auth"))
.map(Cookie::getValue)
.findAny();
Note that it returns an Optional<String> because there may be no Cookie that matches “auth”. If you want to use a default if “auth” is not found you can use:
String auth = cookies.stream()
.filter(c -> c.getName().equals("auth"))
.map(Cookie::getValue)
.findAny().orElse("");