Conversion failed when converting date and/or time from character string while inserting datetime sql server
Bad news: social security number isn't in the format of DDMMYYXXXX. (My own SSN starts with a number above 3, for example.)

Take out the DATEDIFF and CONVERT functions, and just look at the source data. You'll likely see the offending rows just by eyeballing it:

SELECT Users.Id, Users.FirstName + ' ' + Users.LastName AS Medarbajder,
Users.SSN AS CPRNR, 
SupposedlyDate AS STUFF(STUFF(LEFT(Users.SSN, 6), 3, 0, '.'), 6, 0, '.'),
FROM dbo.Paychecks, dbo.Users
WHERE Users.CustomerId=214 AND Users.Id=Paychecks.UserId 
order by 4;

I don't know where you got that formula from, but it doesn't work. (It certainly doesn't produce a valid date for my own SSN.) There are ways to guess approximate dates from social security numbers, but not produce valid birth dates.