Most formally, "Julio and I." If it was just myself, it would be "I was down by the schoolyard," and adding Julio to the party does not change the Ich to mich -- so "Julio and I were down by the schoolyard. (If the enumeration was taking place after the verb, it becomes "the guys down by the schoolyard were Julio and me.")

Most of the above debate is over whether this most formal rule is changing in common usage under certain circumstances. The consensus seems to be that I see it having changed more than most people here, and I'm starting to wonder if this is a regional usage.

---L.