While I don't think that Eggman has ever had an explicit policy along the lines of "liberation of women means the liberation of all women irrespective of their body parts", I think there is some evidence in favor of a positive outlook here.

The thing to understand is that Eggman does not spend very much time outside of his machines. The few times that we do see Eggman participating in social activities outside of his machine, it's typically sporting events, and those sporting events are intentionally of mixed gender -- see, for example, the Olympic Games series. Certainly one of the main arguments used to justify transphobia as public policy would not apply to Eggman; neither he nor his peers have any interest in it.

And of course "Eggman does not spend very much time outside of his machines" carries with it an interesting implication: the machines are the main way with which Eggman express himself to the world, how he interacts with it, and, potentially, for at least some of the craft he uses most commonly (i.e., the eggmobile) he is so experienced that it may now be directly tied to his proprioception -- he is so attached to the egg-o-matic that it may as well be part of his body.

Now, barring some particularly goofy gag comics of questionable canonicity, we can say confidently that Eggman himself was not born with the egg-o-matic, so it is instead a device that became part of him over time through his own effort and volition. This is not to say that Eggman is trans per se but that his relationship to his machines might be analogous to how therian people conceptualize their identity. I'm certainly no expert on that matter, but would feel more comfortable saying that the notion of being therian is related to but not directly tied to gender. This will, perhaps, be one for people to cime in over in the comments.

What I do think this means is that Eggman's thoughts on trans people would mirror that of, say, John Carpenter or David Cronenberg, in that the goals of trans people and the means by which they are achieved are so closely related to his own work that to deny trans people their rights would be, indirectly, Eggman denying himself.

And if there's one thing that any power-hungry dictator who's committed to his craft would never do, it would be denying himself.