This step will probably vary a bit from OS to OS. On OSX it's possible to open files with Preview and save them as PDFs that can be opened with Inkscape. Don't save as PNG, it won't work as well, you definitely want a vector format.
I believe LaTEX has a pstopdf utility that will do this on Linux, please comment if this doesn't work or there's a better way.
Now start Inkscape and open the PDF file. I set the precision to "very fine" but this may not be necessary. Set the units to mm or in, and verify that the dimensions are the same as your Eagle file's.
To convert to DXF, select all (ctrl-A) and object->ungroup. You may need to do this twice, make sure you see "nnn objects of type Path in layer …" on the bottom.
Now select the Node editing tool and hit ctrl-A again, the whole thing should light up with selected nodes. Click the "Make selected segments lines" to convert them to straight lines.
Save your file as DXF format. You can now close Inkscape, optionally saving your file in native svg format.