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Thanks for the information about the SSL! I do use ssl so that will be helpful in the future.
The problem was with the server parameter in the connection options. "T" didn't work but I tried replacing that with "localhost" and now I can connect.
I replaced the .jar postgresql driver in the OpenJUMP Library with the new one, but it still won't connect to my database.
I'm using postgresql 9.5 and OpenJUMP 1.11.

I can connect to the database in QGIS just fine so I'm certain the issue lies somewhere with OpenJUMP but I can't figure it out.