I don't have an explanation (not knowing very well what happens inside the Garch or gig/GARCH routines), but it is not strictly true that it is the same specification: In gretl you used the default "sandwich" for the VCV, and Eviews says "outer product of gradients". You could try to use OPG in gretl as well and tell us what happens.
And using Eviews with exactly the same specification and data I get this: