The circle (sun/moon) design is cleaner overall and simpler – more memorable maybe. I like the square design too but can't help feeling the circle is more eye catching looking at them side by side.
I'm thinking in all applications, web, social media etc, the circle might be more dynamic.
I could put together one more option (hopefully not to complicate things!) with the 3 birds on 3 wires in the circle.
I hadn't actually looked at that. It might not work but might be worth a look just to compare. What do you think?