Can the Trader destroy a disease card?
The publisher's representative (Ryan Metzler) states in this thread that trashing a disease with a trader is an illegal move:
http://boardgamegeek.com/thread/622651/village-trader-question

The rest of it sort of falls apart because he missed that they were erroneously talking about gold values, not cost, but it's the closest thing to an official answer I could find that you can't trash a card with no-cost to the Trader.

Also, clearly stated here:
http://www.boardgamegeek.com/article/7763776#7763776