I think you're exaggerating the importance of Zheng He's expeditions. They were not expanding the Chinese Empire at all, which really wasn't into expansion. They were basically curiosity-driven voyages, designed to collect assurances from the states in the region that they acknowledged Chinese hegemony.
You did not see any actual Chinese control over the said states, nor did you see boatloads of Chinese colonists ready to set up Chinese versions of Jamestown and Plymouth Rock around the Indian Ocean.
Thats the point of a Hegmonic Empire, the native government paying tribute to the foreign government. The basis of it is to have an aknowledged overlordship or aknowledged supremacy and to be able to extert influence on another country. The expeditions were to ensure and more or less coerce powers visited with an awesome display of the Ming's might. Which was certainly achieved for a time.
Actually, you did with several "Chinatowns" set up by Zheng He on his expeditions to these countries-and certainly they would be able to influence local politics with a sizable population of Chinese much like we see with Singapore.
Thats because the visited areas by Zheng He were already densely populated and governed by some form of political government.
I certainly could see more or a Jamestown rather then a Plymouth coming out in the Americas. If not then its possible that it could be an option for Merchants, Reformists, refugees (fleeing Famine) and etc fleeing oppression and the natural disasters that rocked China in the Ming time period. If the Americas were discovered and word of a primative sort of people beyond any local groups maybe Chinese or Japanese pirates could move in....
Though I stand that if the Americas were discovered even if the Chinese Imperial government was not interested on a commoner level it would drawn interests. For Jade or SIlver seekers porbably least likely but mostly for fishermen (The Pacific Northwest area is a overly, overly, overly abundant area of 'natural resources'). You would see not long Junks sailing up with Native American Canoes for fishing and some trade. It would also be logical for any fishermen (or whoever else) were to go to the area for a long time period would set up a semi-perm. settlement along the west coast.
I point out again during this time period, Japan is going through its Sengoku Period, which was also a time marked by a rise in Japanese Piracy and most likely refuggees fleeing the violence (wheather they be just common people or some noble house that has had its power crushed during the events of the civil war) who may look to a bountiful place as a good area to at least wait out the violence and live in peace or exile.
Taking into account Refugees from China and Japan the most ideal place would be the Chinatowns set up by Zheng He or the newly discovered and lax controlled American West Coast becuase those would be the only places to go. Japanese refugees would not be welcomed in China or Korea and vice versa for Chinese refugees. The best palces to go would be the 'Chinatowns', but there they also could be mistreated by the local majority populance and rulers (Especially if they were some group like the White Lotus, who could feel betrayed by the founder of the Ming dynasty for throwing off his beliefs to gain favor with the Confucians) so the best place would be a place without any authority. The Americas.
Hmmm The White Lotus could found a 'Plymouth' on the west coast....possible timeline there....