I figured it out! Even though the app had a hard time telling me what the working directory is, I managed to figure out that it is the folder that the application is in. Then I was able to solve it by referring to the app as a directory:
new File("MyApp.app/Contents/Resources/MyFile.txt").exists() //Returns true!!!
I'd also like to add that I found the suggestion on this blog post to add the following to my info.plist file:
<key>WorkingDirectory</key>
<string>$APP_PACKAGE/Contents/Resources</string>
Unfortunately, this doesn't seem to work on Mac OS 10.7.5 and the working directory just gets reset to the folder that the application is in.