Hi busters dad, and welcome to the forum
It's going to be hard to answer your question without knowing which Linux distribution and version is on the ASUS netbook.
I can tell you that if necessary, and you're willing to update the Linux distribution that's installed that both of those printers are supported .. in fact the HP printer will be supported "out of the box" as most Linux distributions come with the hplip drivers pre-installed
Most likely the Linux distribution on the ASUS wil already have hplip installed, but it may not be a recent enough version (at least version 3.13.8 of hplip) to support that particular printer.
It would probably be best to come back when you have the netbook so we can discover which Linux distribution/version is on n it.