Same thing you do to an injured person who doesn’t want medical help. Let 'em die.
But really, do they have a particular reason for wanting to stay with that hosting? I’ve only had one client who had windows hosting—simply because he didn’t know what he was doing. I just told him how crap it was, and he switched to Linux the same day.
The thing to remember is that GoDaddy is in the domain registration business - their hosting is just an add-on service that they don’t put much effore into as idoing so would take away from their main business.
You would get better web hosting from someone where hosting is their main business instead of just an add-on to try to make a little extra money out of the domain name sale.
It seems they are applying their own rules for that kind of web hosting service and they might not appear as traditional one. I would recommend OP check other web hosting companies offering that and most likely better and acceptable alternative will be found.
It really depends on the situation. If they only use Windows technology like .NET or silverlight, you’d better go for Windows hosting.
.NET does run under Linux but many libaries and addons or plugins created by third parties use libraries that can only be found under Windows and won’t run under Linux. So Windows would be a safer option.
Still, it may be a good excuse to renew their website into something that can lower the hosting expenses and make them less dependable on the OS that’s used in the hosting server… so it is up to you how you want to sell it