One more update:
The Segger library is closed source, licensed to Cypress without source code. Cypress allows us to use it for free with the emFile component.
It is not ideal for using with the PSoC 5, because you don't have access to the source code. The library does an unhandled exception if a card is removed or not installed during all of the calls that I have tried. At that point, you have no choice but to use a watchdog timer and reboot, or go purchase the source to the library to correctly patch this unhandled exception. The API for segger has no described method of handling this error.
The elm-chan library, on the other hand, is free to use, has the source code, and does not crash and burn if a card is removed in the middle of a process. If it did, then you could patch the source code and fix the issue. The elm-chan library is an MIT style license, you get to use it without revealing your source code. It is used by many other embedded processor developers for other mcu's.
It seems to be working well for us. There is at least one project on github that uses a modified version of the library, for the CY8CKit-059 board. You need to be a bit wary of that project, it appears the author modified some files and seems to have broken code page support. However, it is a very good test case to allow you to check out your hardware before you use your own code.