About six years ago, another pin with a carpenter's square on it showed up. It is also 7/8 inch diameter. This one bears the name Parker. Alton B. Parker was Roosevelt's opponent in the 1904 campaign. The phrase "square deal" has always been attributed to Roosevelt. But Alton Parker apparently used the carpenter's square first, albeit without the slogan. Except for the handful of these pins that appeared in 2002, to my knowledge it has not been seen before or since.
Here is a fourth "square deal" pin. It is also 7/8 inch diameter. I found it on eBay a while back. It is the only one like it that I have seen. The seller probably did not even see the meaning, let alone consider it a campaign item. It can only mean "square deal," and is probably a Theodore Roosevelt campaign pin.
If you have a "square deal" pin that is different from these, please post a comment about it.