Tammy, this is what I got from HP "The drivers for your HP product are already included in the new Windows 7 operating system! You do not need to download drivers to use the product . Simply connect the printer to your computer with a USB cable, and Windows Update automatically installs the driver for your product." This here will tell you how to proceed with the installation of your printer. Hope this helps, good luck.
It will work, however you will lose the full utilities with Windows 7(I use a 842c with Windows 7, so I know this for sure), and you will get reduced tools for the printer
If you want the utilities in Windows 7, you're gonna at least need to download and install the Windows XP drivers, but if they wrote Vista drivers for yours, you would want to use those
Here is the driver(Windows XP)
Here's how you will want to install it if just installing it without doing this fails:
On the download, right click properties and select Compatibility, then check the box that says "run this program in Compatibility Mode" and then select Windows XP, click Apply and it should install
As to the utilities, you will want to do the same thing with them as you did for the driver if you need to do so, or it fights you to install