I'd say that converting your EPS file to PSD is more of a designer job because it involves manipulating graphical files. But then again slicing a PSD can also be considered manipulating a graphical file.
It's kind of a blurry border here, but since Illustrator is build specifically for vectorial design and vectorial file manipulation, I'd say the task should be assigned to a designer rather then to a front-end developer.
I don't think I've heard many developers having Illustrator as part of their "arsenal" but it's always a good ideea to have it installed if the need arises :)
Long story short: Yes, the task should be assigned to a designer.