Sorry fkdiver, I just noticed your question. If you haven't gotten a response elsewhere, the solution is fairly simple.
In a text file prepared as input to Across Lite (in order to create a .puz file, solve the puzzle using the program, or print the puzzle) the program can handle apostrophes and quotation marks in clues just fine but with one limitation: these marks need to be simple/straight (' or ") characters. So-called "smart quotes" in clues will result in the behavior you've observed. (These are apostrophes and quotation marks that don't point directly up/down; they are generated for example by MS Word when turned on in Settings > AutoCorrect).
So if you turn off smart quotes or work from a very basic text editor to start with, your apostrophes and quotation marks will render just fine in Across Lite.
Hope this helps!
