That's their problem, not yours. As long as you wear shoes in the lav, you're not going to get their foot cooties unless you return to your seat, take off your shoes, and put your feet where the bottoms of your shoes had been.

As for the security screen: Disposable foot covers are fairly cheap (cheapest I saw on the web for antiskids were $10 per 100, incl shipping, but since they were meant to fit over shoes they might be too big for some). Disposable booties (as seen in hospitals and first class amenity kits) seem to be more expensive. Target.com has expensive ones specifically for this purpose:
"• Stay clean while traveling with FeetWrap Disposable Slippers
• Slip on to protect feet when taking shoes off during airport security checks
• Avoid contact with heavily trodden airport pathways that can be breeding grounds for foot fungi"