That answer there is just wrong. It represents that the Lubavitcher Rebbe says that the level of Neshamah is not present with non-Jews. In fact, the Lubavitcher Rebbe says that all 5 levels of Nefesh, Ruach, Neshama, Chaya and Yechida exist in all human souls. The distinction between G-dly and not is precisely in the additional soul (which is what Tanya speaks about - a whole separate one - in other words a divine connection and influence necessitated and determined by the additional Mitzvos required of a Jew over a non-Jew, and the corresponding ability to actualize those requirements).
In other places it speaks about the language of Ger Shenisgair - a convert who converts - rather than a non-Jew who converts, that actually the potential for conversion was pre-existing (he was a convert before he actually converted). Thus that very same influence and connection can exist with someone who is currently non-Jewish. My own observation is that this influence is often seen in many conversion stories - a pull and attraction to Judaism that just seems to come out of nowhere.
I think the misunderstanding comes from the fact that Neshamah (in the Kabbalistic system) refers primarily to the intellectual capacity of the soul, and the G-dly soul is described as primarily intellectually motivated, thus Neshamah can be thought of as being more associated with the G-dly soul. But that is a general association (the Neshamah aspect is more predominant in one over the other), not that it doesn't have it.