Dropbox has version control (you can restore old versions of a file) and does not seem like a terrible solution to the problem. I keep the Eclipse repository in Dropbox, so it is available on any computer; but since I use it myself, I have not come across your problems.
There is one case where I see how you ran into problems - this is if your class files are also stored in a folder. It just ruins everything. Make sure you specify the location on your local hard drive for all build artifacts (classes, banks, ...) and that the only things in your Dropbox are .java sources.
In fact, I suggest you not save the eclipse project in your Dropbox, just create an eclipse project and point it to java files in your Dropbox.
If this does not work for you, go with what other people have said here and create an SVN repository somewhere, it's easier than you think.
Oh, another possible problem is dates! You may want to make sure that the date in your java files does not skip back and forth (as can happen if one of your developers was in a different time zone). In this case, Eclipse may not recompile your file.
Also, instead of copying / etc, which you are currently using, try to make the project clean.
Response for requesting additional information:
When you start Eclipse, select / create a workspace that is NOT in your Dropbox. The best place is probably outside your home directory. If you have already specified the default workspace, there should be an element of the switch workspace in the file menu.
Create your own project. select "Create a project from an existing source" and specify the source files in your Dropbox. I think you want to “create separate folders for source and class files” to save class files from your Dropbox. If you see something saying “Copy files to workspace, say no.
This should give you an ongoing working draft. I hope you no longer see these problems.
One more thing can help - and it can work on your existing project - without the above procedure ...
When you update your files (f5) to download changes from Dropbox, select the Project / clean menu and select the project. This should remove all class files and rebuild them.
If your class files are shared on Dropbox, this can still have strange consequences for other people with open eclipse, so I really suggest rebuilding the workspace, as I said above.