@Nishad had the right idea, but it turns out that something else needed fixing as well. For the sake of being thorough, here's the final batch file code I wound up using:
SET /P Param1 = Enter job number here:
SET /P Param2 = Enter current status here:
sqlcmd -S DevTest -d test_database -Q "EXEC dbo.JStatUpd @jobNum=N'%Param1%', @stat=%Param2%"
PAUSE
It was only with this specific formatting that I could actually get my user-entered values to work properly; @echo showed me that I was getting blanks no matter what I entered, so I started playing with the query parameter formats and found that this worked.