"So my question is... will a 32 bit program run on windows 7 HOME PREMIUM 64-bit?"
Yes, all W7 versions are able to do so.
Whether it's compatible with W7 you can check compatability by running the Upgrade adviser for both hardware and software.
It's advisable to do this prior to upgrading.
All 64-bit editions of Windows can run 32-bit programs, not just "Ultimate" or "Pro"
What "Ultimate" and "Pro" can do is to run a copy of Windows XP in a virtual machine that you can use to run 32-bit programs which, for one reason or another, don't run properly under a 64-bit operating system. 32-bit programs with that problem are relatively few and far between.