For a typical first floor load a 6x6 in eastern white pine can span 11' at 24" o.c. The reason why they are rarely used this way is because a rectangular section is much more efficient in a simple load bearing beam. For example 6x6 will fail in bending over a 12' span, but a 4x8 will suffice even though its cross section is 4 sq. inches smaller! (although deflection is a little high).
But perhaps the best reason not to use 6x6's as joists is that they depart from the golden section, unlike the beauty of a 4x6.5, 5x8, or 6x10. And who can argue with fibonaci?
oh and or course, it is a really lovely barn already....
