It's standard usage to say "a white man's country", whereas it would be unusual to say "a white men's country", since the expectation is that an indefinite article is followed by a singular noun.
In this particular context there's a further complication in that it's ambiguous whether the indefinite article relates to "man" (in other words, it's a country belonging to the white man) or to "country" (in which "white man's" or "white men's" acts as adjectival phrases). In the former construction, "a white men's" would be ungrammatical, whereas in the latter construction it would be perfectly fine grammatically. Such ambiguity is sometimes avoided by hyphenating the words in the adjectival phrase: "a white-man's country" or "a white-men's country".
Note that while the non-gendered expression "a white person's country" is often preferable as it's more inclusive, in this particular historical context the gendered usage could be seen as appropriate.
It would be grammatically correct to delete the indefinite article and use the plural - "schemes that aimed to make the U.S. white men’s country" - but this is not a common construction. If you were intent on using the plural men, "schemes that aimed to make the U.S. a country for white men" would be a suitable alternative.