EDIT: (removed previous reply)
"Max. Attacks: This indicates the maximum number of attacks that the eidolon is allowed to possess at the given level. If the eidolon is at its maximum, it cannot take evolutions that grant additional attacks. Attacks made with weapons, including those granted by a high base attack bonus, are counted against this maximum."
No. It goes even dumber than that.
It doesn't care about source of attack (I have -a- sword), it cares about how many attacks you can make with it (Bab 6 means 2 sword attacks).
So it doesn't matter that you only have one unarmed attack. Your can still TWF with it, and that's an extra attack, and that's all it cares about.