Provided all flights are on the one ticket, UA rules apply for the whole itinerary.

US DOT requires that the baggage allowance for ex-US itineraries is whatever the allowance is of the first "marketing carrier."

If OP was booked as ORD-(LX)-ZRH then LX allowance applies. If it was EWR-(UA)-ORD-(LX)-ZRH then UA international allowance applies.