Στις Ηνωμένες Πολιτείες, εμέις δεν λέμε "A juice." Η λέξη juice έιναι uncountable (αμέτρητος).
I think it's one of those cases where you can say "I'll have a coffee" or "I'll have a juice" meaning "I'll have a cup of coffee" / "I'll have a glass of juice". Would that be appropriate?
Yes, but we most likely say "a glass of juice" (as you did), not just "I'll have a juice."