It's true that when you're checking for exact equality, that order does not matter.
But that's not to say that the order in the index does not matter -- perhaps this is what your co-worker was trying to say. For example, if I have a table:
PersonID FName LName
-------- ------- -----
1 John Smith
2 Bill Jones
3 Frank Smith
4 Jane Jackson
...
(assume a significantly large table)
and I define an index on it in the order (LName, FName), that index will necessarily perform differently than an index defined in the order (FName, LName), depending on what the query is.
For example, for the query:
SELECT * FROM People WHERE LName = 'Smith'
, you will most likely get a better plan for the first type of index than for the second type.
Likewise,
SELECT * FROM People WHERE FName = 'John'
will perform better with the second index structure over the first.
And
SELECT * FROM People WHERE FName = 'John' AND LName = 'Smith'
will perform identically no matter what order the index is created.