If they insist on using Linux and you insist on using Windows, then they may insist on using someone else for their implementation.

The best you can do is communicate with them the reasons you think Windows is a better platform for this particular solution while trying to draw out the reasons they want Linux.  If the reasoning behind why they want to stay with Linux seams irrational to you, it's probably because they aren't telling you all the real reasons.  It may be a pricing issue.  It may be that their people are already comfortable with a linux environment.  It may be that the owner's son only knows linux and doesn't want to have to learn something else... lots of possibilities.  I worked for a financial company that used Mac's entirely because the owner of the company tried to sell Bill Gates on financial services (estate planning specifically) and Bill Gates didn't take him up on it. 

Ultimately you have to either convince them to change, or decide if it's worth it to you to do an implementation in Linux.  If so, Linux it is.  If not, tell them to find someone else who is willing.