To be honest with you it does say on the avermedia site the format should be NTFS ideally but I am not sure why using this format seems to be a problem for you? If you stuck to FAT32 you may have got outages as the max file size is about 4Gb, so as your recording hit this limit it would be unable to continue further with that file.
You say "It doesn't have an internal HDD and no one (not even Avermedia) sell one specific to this product" but it does not need a drive specific to the device. For internal fitting it uses a 2.5inch SATA drive which is a widely and very commonly available type of HDD. Caution here though because the GCHD itself can get hot and the internal space for the HDD is not actively ventilated. I know this makes no odds to you but may be of interest to others. One review I read said the temp reached over 80 degrees C (think it was actually 90 degrees C) - bad news for a HDD then. Attaching a HDD as an external device is via the universal USB - again a widely used format.
If you are having to attach an external HDD you may as well attach a laptop and therefore the Hauppauge HD PVR comes into play but don't expect much in the way of support even though they do have a UK presence. I could not join their forums as my email client (Gmail) was banned (!) and my questions I therefore had to ask via email have never been answered.