Sargability of queries against a view with a case expression
Is it possible to still have my ShipmentId output go through the case
  expression, but also have a sargable query when ShipmentId is used in
  the where clause?


No.  SQL Server doesn't do that for you.  

This is where you might use an inline Table-Valued Function, sometimes called a "parameterized view".  EG

CREATE function GetShipments ( @ShipmentId int )
returns table
AS return

    SELECT  CASE 
                WHEN shipmentOld.SHIPMENT_ID = 0 OR shipmentOld.SHIPMENT_ID = -1 THEN NULL
                ELSE shipmentOld.SHIPMENT_ID
            END AS ShipmentId, 

            OtherValue, SomeOtherValue, OtherStuff
    FROM    dbo.tblShipment_Old shipmentOld
    where  SHIPMENT_ID = @ShipmentID


Or add an extra column to your view.