I have two columns that are joined together on certain criteria, but I would also like to check if two other columns are identical and then return a bit field if they are.>
What's wrong with CASE for this? In order to see the result, you'll need at least a byte, and that's what you get with a single character.
CASE WHEN COLUMN1 = COLUMN2 THEN '1' ELSE '0' END AS MyDesiredResult
should work fine, and for all intents and purposes accomplishes the same thing as using a bit field.