No manufacturer has announced a free upgrade program for systems running Vista Basic. All the offers are for Home Premium/Business/Ultimate. HP and other manufacturers aren't obligated to even offer these free upgrades. We expect them and welcome them,but there is no obligation.
I'm a systems technician for a large company and have been playing with the Windows 7 RC for some time. Our enterprise agreements allowed us access to the MSDN betas also. We've been avoiding Vista in our company but look forward to Windows 7. Sadly there is no direct upgrade path from XP to 7, it does a parallel install, moving the previous OS and files into a Windows.OLD folder on the drive. All the drivers and applications have to be reinstalled after this. For any machine running any version of Vista it is possible to fire the DVD in the drive and have it upgrade to Windows 7.
TLDR: Yes, Microsoft Windows 7 supports a direct upgrade from any existing Vista version to Windows 7. HP's (and Toshiba, Sony, Lenovo...) free upgrade program does not extend to systems that ship with Home Basic.