After a quick e-mail to the folks at JNIWrapper, I found out how to do this. I was missing the Pointer wrappers around strings in the structure and I didn't pack the structure. Once I did that, it worked! It even supports the callbacks!

Here's the code in case you're interested: 
\nUShortInt version = new UShortInt(7000);\nAnsiString output_dir = new AnsiString("c:\\tmp");\nconnectparms = new AnsiString("uid=<...>;pwd=<...>;dsn=<...>");\nAnsiString startline = new AnsiString("");\nConfirmCallback confirmrtn = new ConfirmCallback();\nErrorCallback errorrtn = new ErrorCallback();\nMessageCallback msgrtn = new MessageCallback();\nStatusCallback statusrtn = new StatusCallback();\nChar bitFields = new Char((char) 15); // flags are 00001111\nAnsiString hotlog_filename = new AnsiString();\nChar backup_interrupted = new Char();\n\nStructure backup_info = new Structure(new Parameter[] {\n   version,\n   new Pointer(output_dir),\n   new Pointer(connectparms),\n   new Pointer(startline),\n   confirmrtn,\n   errorrtn,\n   msgrtn,\n   statusrtn,\n   bitFields,\n   new Pointer(hotlog_filename),\n   backup_interrupted }, (short) 1);\n\nErrorCallback dbtinfo_errorrtn = new ErrorCallback();\nStructure dbtinfo = new Structure(new Parameter[] {dbtinfo_errorrtn});\n\nLibrary dbtool = new Library("dbtool7.dll");        \nFunction dbbackup = dbtool.getFunction("_DBBackup@4");\nFunction dbtoolsinit = dbtool.getFunction("_DBToolsInit@4");\nFunction dbtoolsfini = dbtool.getFunction("_DBToolsFini@4");\n\t\t\ndbtoolsinit.invoke(null, new Pointer(dbtinfo));\ndbbackup.invoke(null, new Pointer(backup_info));\ndbtoolsfini.invoke(null, new Pointer(dbtinfo));\n\nconfirmrtn.dispose();\nerrorrtn.dispose();\nmsgrtn.dispose();\nstatusrtn.dispose();\ndbtinfo_errorrtn.dispose();
And here is a sample callback (they are all the identical with different class names): 
\n/*\n extern short _callback ErrorCallBack(char far * str);\n*/\nclass ErrorCallback extends Callback {\n   private ShortInt result = new ShortInt();\n   private AnsiString message = new AnsiString();\n   public ErrorCallback() {\n      init(new Parameter[] { new Pointer(message) }, result);\n   }\n   public void callback() {\n      System.out.println("ErrorCallback -> " + message);\n      result.setValue(0);\n   }\n}
This means the only JNI I need in my application is the PowerBuilder to Java bridge. I just got rid of a whole bunch of C code with this thing!