Why Scotty should have found it shameful for a Scotsman not to be "up on Milton" was unclear. His country and England were not merged into the kingdom of Great Britain until the 18th century, and Milton was – in every sense of the word – an English poet. If not nationalist, then, Scotty's embarrassment may be religious: Milton was strongly influenced by Calvinism in his years over on the European mainland, and Calvinism was a prime ingredient in Presbyterianism, a powerful force in Scotland's culture.
However, this was most likely a plot device to give Kirk a reason to explain the comment, as it was very likely that the audience would not be familiar with the quote from Paradise Lost.
According to the Star Trek Encyclopedia (3rd ed., pp. 302-3}, the birth and death years of John Milton were 1608 and 1674, respectively.