Okay, this is the thing. I will try to explain this the best I can before I go to sleep.
Feminism is usually regarded as being left wing, democratic liberals. You can fill in the rest of the blanks.
Egalitarian is libertarianism which is on the FAR right of the spectrum, past conservatives, but has an entirely different meaning than a conservative republican.
Confusing? Yes, and I can see why.
However, egalitarian deals with a wider variety of issues that neither repubs or dems in America take part in. So if you are going to refer to yourself as an Egalitarian you will have to refer to yourself as an independent. Meaning you don't take sides.