Thank you very much for the original GraphML file.
You are using lots of symbols with semi-transparent colors (i.e. all those gateway symbols form the BPMN palette). Unfortunately, EMF does not support transparency (as far as I know that is a restriction of the format and not a shortcoming of yEd). Either use only opaque colors or choose an export format that does support transparency such as PDF or SVG.
Moreover, yEd does not (fully) support exporting SVG symbols (such as the display symbol from the Computer Network palette) to EMF. Either convert your SVG symbols to PNG and use those or (once again) export to PDF or SVG.
Here is your sample file modified according to the above advice. Try exporting that version to EMF.
It is possible to convert SVG symbols to PNG using yEd. To do so, create a new diagram with the SVG symbol as the single node. Then export the diagram as PNG. Make sure to choose a small "Margin" and "Transparent Background" from the PNG export settings. Afterwards, import the PNG symbols as described in How can I use my own node symbols in yEd?