Stop using EPS or any other form of postscript.
Save an Illustrator file as AI or as PDF/X-4 and place that.
EPS is an old, outdated and lossy file type as it does neither support transparency nor color management and has several other issues. EPS is a file type which was never further improved since the late 90s.
Where Willi's advice is sound, this should still not be happening, even using EPS.
There's something in the artwork causing those parts to Knockout - I don't think saving to a different format will resolve that.
Check your artwork for transparencies, spot colours, and how those white areas are set, if set to overprint, knockout etc.
We need a little more info.
Sometimes, when I have no time to investigate, I just select problem object (Tasmania, New Zealand, etc. in your case) and just copy ALL attributes from an identical but "working" area, let's say, from Australia. This way you sometimes fix the problem even without knowing what it was...
Managed to solve the issue somehow.
Had to flatten transparency although this didn't appear previously in the previous files.
You should NEVER flatten transparency, this is done anyway with saving as EPS. You should change your workflow and work with AI and PDF instead of EPS.