Straight out of the box, I don't think the FlexViewer can do that, but I'm guessing it could be done with a bit of coding (Though I'm not a Flex programmer - Robert?)... But it also seems like if you group your operation layers properly in a service, you could have all the layers the user might want without too much clutter, and you give them control. If you really want to preset what operational layers turn on for a give base map, it is also easy to clone your site and just change out the operational layers, and then have the user just choose which site to enter... Not the answer you were looking for, I know, sorry.
