Long story short,
.eps is the slow cousin of
.pdf if possible.
.pdf problem is, I think, no problem at all, as long as you save your file as Pdf 1.4 or higher, you should be good to go, because it is the earliest pdf version to support transparency.
Since you said that
"it always comes with a full white background", I'm thinking that if you viewed the
.pdf in Acrobat or actually almost any application that takes in PDF, it will have a white background for convenience. Imagine reading text documents with a transparency grid on.
Acrobat and many other applications have what is in Adobe applications called a "transparency grid", which helps you view the document without the default background. It's not always called that in every application. In Acrobat:
settings (Cmd+K) > Page display > Transparency grid. Photoshop and Illustrator also have a transparency grid.
You could also just open the output image back in Illustrator to make sure nothing changed since the export.