Is there any possible way to have a well-developed american nation and have such a nation own a colony in Africa or Asia like the U.S.A. had with Liberia?
Liberia was'nt really a Colony...
Let's not confuse labels with reality. Liberia was a colony and was repeatedly referred to as such by people at the time of it's founding and for decades afterward.
The terms "colony" and "colonization" are very broad and cover a multitude of of historical and current situations. In 2010 places as diversely governed and administrated as Bermuda, Gibraltar, Guam, and Samoa are still considered colonies.
While that may be, from the OP's question it seems more he's referring to Colonies in the manner in which most existed in the 19th century, either as unrepresented territories milked for their resources and economic benefit to the Home country, with little to no investment into them or areas that a country had decided to outright annex and incorporate into itself.
... like the U.S.A. had with Liberia.
Liberia was neither of these, hence America never had African colonies in the above sense.
From the OP's question, he's specifically including colonies like Liberia.
If you don't want to believe me, re-read the last six words in his original post:
Why shouldn't we take the OP at his own words?
In your sense perhaps, but definitely in the sense of the OP's question because he mentions Liberia as a specific example.
I think we're going by different ideas what he meant.
While you're going by what he said literally...
... I'm assuming he does'nt realize that Liberia was'nt a Colony in the sense I described.