Its true that Charles - as both Kind of Spain and Emperor - had both symbols in his personal arms. There is even a small Spanish town that uses these, in reference to Charles personal arms, but that has no bearing on their usage by 16th century units.
In my research I never found a depiction of pillars (or any other Spanish emblem) on any Landsknechts flag, nor can I imagine that any Spanish unit would ever bear the Imperial eagle. Both were allied by their allegiance to Charles, but apart from that there was a lot of ongoing rivalry with several bloody incidents.
I would be (really) glad on any information to the contrary (regarding arms and emblems), though.
Alas, that flag even made it into the plastic set of Warlord Games. Which, on the other side, kind of fits...