I am trying to export a logo I have created but it is off centre, I can't figure out how to fix it

When you save as EPS, unless you check the option to "Use Artboards", only the area surrounding all the objects is defined. I'm suspecting your type on a path is still live text, and as such requires an extra amount of bounding box (for technical reasons I won't get into) so it's showing extra room at top and to the right "to be on the safe side".

For your final logo files, you should convert your type to outlines anyway, which would eliminate that extra bounding box issue, but you still need "something" to keep the logo centered: Alternative to selecting "Use Artboards" You could also define your own rectangle to wrap the objects to keep the centered orientation, with no fill and no stroke, and then make it a clipping mask, which will essentuially crop the logo as you wish. In this case you don't need to select "Use Artboards".

What I do... is I select everything in Illustrator.  Then I copy it.  Then I go to Photoshop and I hit paste. I then choose to input it as a smart object.  Then I center and place it the way I want into the blank Photoshop document and then save as a PNG.  This is so much easier and there is no error.

Your screen shot with the 4 logos shows the .ai files correctly centered and the .eps files positioned at the bottom left, is this correct?

How are you viewing the .eps files? Try saving the .ai logo as an Illustrator PDF, using the Illustrator defaults.

So this happens when you export it as ... a PNG?

In that case check the option "use artboards"