Windows Small Business Server 2003 SP2 upgrade plan
Solution 1:
No, you shouldn't insist. It's not your decision to make. It's their decision to make and it's your job to advise them on making a decision that works for them and to advise them of the risks of continuing to use an OS without official support, if that's what they decide to do.
Opinion/Rant: The idea that because official support is ending means you need to replace or upgrade what you currently have is ill-conceived. This idea is counterproductive (IMO) and I see too much of it in the IT realm. The end of official support doesn't mean that the product will stop working.
Not having official support and no longer receiving updates and security patches is, without a doubt, a serious concern but it doesn't mean that you have to scrap what you have, especially if you're in a financially sensitive situation. If you've advised them of the pros and cons of continuing beyond the EOS date and the organization understands the ramifications and is OK with that then you've done your job correctly. There are plenty of organizations still running Windows 2000 and Windows NT (shudder) and are perfectly happy doing so. There are cases where moving to a newer OS just isn't feasible or possible.
This "blanket" idea that you need to upgrade because of EOS is the product of a "sheep" mentality that presumes that every organization fits into the same neat little box. That viewpoint doesn't consider the customer carefully enough. It's not my job to ram technology down my customers throats. Its my job to advise them on the best way forward after considering their needs, requirements and limitations.
Solution 2:
There is no upgrade path from a 32 bit Windows OS to a 64 bit OS, so you'll be reinstalling/replacing rather than upgrading.
And I wouldn't be insistent on an upgrade to 2012, necessarily, as they could probably get more value out of SBS 2011 or even Server Essentials 2012 (but SBS 2011 is probably a better choice -as much as I loath the SBS line, it does provide a lot of otherwise expensive enterprise apps for cheap).
As to whether to upgrade or stay with what they have, well... business needs drive IT, not the other way around... so there may well be a strong argument for staying with SBS 2003. Having said that, however, some components of SBS 2003 are going out of support this year. (SQL 2000 in April and Sharepoint 2.0 in July.) If supportability is important, you'll need to upgrade sooner, rather than later.
And one final note as you ponder whether to upgrade or not - Microsoft killed off the SBS line and is currently pushing for small and medium organizations to move to its cloud. Cloud services like Google Apps and Office 365 and hosted Exchange (etc) might be a cheaper and/or better option than investing in a new server and the OS and applications to support the organization. (I'm not a big fan of the cloud either, but sometimes it is the better option - something to consider.)