Can crossing your legs while sitting cause neuropathy and varicose veins?

Possible: If you maintained the position for hours, you might compress the peroneal nerve at the lateral fibular head, which could result in a foot drop, but you would most likely suffer from a genetic syndrome termed "hereditary susceptibility to pressure palsy". I would doubt that varicose veins would ever be caused by crossing your legs.

Unlikely.: I doubt that crossing of the legs will lead to either neuropathy or varicose veins. For neuropathy to occur, it would mean that there was pressure on a nerve to cause the problem. This would be quite difficult to happen with simple leg crossing. Varicose veins are due to malfunctioning valves and crossing of the legs will not damage the valves although it could decrease the venous return.

No: Exercise is good for your legs but that doesn't mean sitting with your legs crossed creates issues.