That is very strange, I can confirm what BE1 said, nothing is actually installed (as in written to the registry) when you install the programs they are just copied to the directory of your choice.
Going by what you posted it looks like something is triggering one of the error log files to be opened when you boot , the Files (x86)\HG612_Modem_Stats\Ongoing_Stats\ERROR.LOG_file_ERROR.TXT text looks like part of the directory name, I say part because one would expect it to be Program Files (x86). This is not stored in the log files, so no idea where that is coming from.
Did you copy the program files into the Programs Files (x86) folder?
You can also look in System Configuration to see what is starting at system startup, press the start button and type System Config and you should see it appear in the list, select it then select Startup and see what is listed.
Also try deleting the HG612_Modem_Stats folder and it's contents, as you haven't logged any stats there's nothing to be lost. Also ensure the scheduled task is deleted, either from the GUI tasks menu, or directly from Windows 7 Task Scheduler.