Rather than simply testing for a question mark followed by a character that is not a quote (\?[^"]), I should have tested for a question mark with a negative look-ahead (\?(?!")) for a quote. This appears to work:

$myvar =~ /"((?:[^?"]|\?(?!"))*)\?"/';
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For those unfamiliar with lookaheads, they allow you to test for text without "bumping along" the regex. In other words, \?[^"] will check for a question mark followed by a non-quote character, but further matching of the regex continues after the non-quote character. \?(?!") allows you to check for a question mark not followed by a quote, but continues matching after the question mark.

Note: There is a subtle difference between the negated character class and the negative lookahead. The negated character class generally requires a character after the question mark (in the above example), while the negative lookahead just makes sure that a quote doesn't follow the question mark and doesn't require a character.

Cheers,
Ovid