Connections can export custom fields without changing the code in Connections itself, but, that said, one still must hook into the export process to export the custom field data. The reason being the custom fields can contain any arbitrary data so they is no way to know how to process the data to export it.
Now, if what you want to export is just a text field, the code to export it is really simple. Assuming this is basically a continuation of this thread, I can spend some time this week to put together two tutorials to compliment it, one for CSV Import and one for CSV Export.
Let me know.