In 1868 I don’t think there was any such thing as an illegal alien was there?
I'm not sure what the genesis of that particular phrase is, or when it was coined, but the Chinese Exclusion Act of 1882 predates US v. Ark by about 18 years or so. So, at the time Ark was heard & decided, there were classes of people that were disallowed entry into the US.
If there were Chinese who did enter anyways, I'm not sure what if anything was done to them. I don't know if any kind of deportation proceeding had yet been established.