So I figured out the answer.
First of all, you can get a better idea of how FlexBuilder compiles your app by adding a -dump-config=C:\myConfig.xml to the compile arguments in FlexBuilder. This outputs an xml file containing configuration settings used in the compilation step. You can also use this file as an argument to compc or mxmlc if you'd like. Read more about it
here...
But, here's what actually solved my problem. I was using the regular old Flex SDK installed on our integration server to compile my apps using Ant. This is the free SDK you can download from Adobe's site. I then took the FlexBuilder directory from my local machine and copied it up to the integration server and pointed my build script to use that version of the SDK (and changed my path env variable).
When I compiled using the FlexBuilder version of the SDK, all was well and the strange bugs I was seeing in my app disappeared.
Moral of the story, make sure you are using the same version of the SDK for your automated builds as you use to build locally.