Hi Aaron, I've attached an edited version of your model where I've sliced off that little bit I showed above and that seems to have fixed it up ok.
The process I used was to select just those immediate surrounding faces and use Edit > Separate to break those off of the main object. Then I drew in a cutting line like I showed above and used Edit > Trim to cut that with the line and discarded that touching piece, then Edit > Join to glue the surfaces back together and then select the object and run Construct > Planar to seal off the planar hole that was left by doing the Trim.
The result is now a solid volume and it should behave more like you were expecting during the booleans.
Basically the booleans look at what volume different pieces are contained inside of in order to determine which things to keep and which to discard. When your object does not enclose a volume and is just an open surface it will just get cut up as a surface as well.
Hope this helps!
Also when you do some more cuts you may need to be a bit careful about where you place the cutting line to avoid the same kind of situation of 2 different things just barely grazing each other at a point making things become self intersecting. In order to have a well defined solid each edge must be joined just between 2 surfaces and in a case like you had originally you've got 4 edges are running over the exact same area.
- Michael
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