What they want is a vector art file("line art"); what you have is a raster image file(pixel-based art)
There is no conversion from bmp to eps per se, since the bmp is composed of a grid of pixels, while an eps is a resolution independant file based on paths, or "lines".
ie: logos begin as eps files and are often rasterized, then converted to jpeg for websites, as an example.
You might be able to load the transparency mask as a selection in Photoshop, then convert it to paths, but this will never be as clean as the original Illustrator file.
Sometimes, a 300dpi DCS 2.0 (Photoshop eps format, not to be confused with the "Illustrator eps" Kinkos is likely referring to) will work fine for color separations, as long as the image is large enough to print clearly.