HP Deskjet 5440

There are two kinds of print driver on OS X - Carbon API and CUPS API. (I'm talking about non-postscript printers now) *
CUPS drivers are required in order to use the standard network protocols, including Windows Printing and IP printing.
Brother is the only non-postscript manufacturer so far to make use of the CUPS API and provide CUPS drivers.

Apple tried to solve this problem by including the Gimp-Print drivers, which are CUPS drivers. But newer HP printers aren't supported by Gimp-Print.

*(postscript PPDs/drivers are automatically CUPS-capable because postscript [& PDF] is the native output of all apps that use the Mac APIs)

Read these:
http://discussions.apple.com/thread.jspa?threadID=558242&tstart=0
http://discussions.apple.com/thread.jspa?messageID=2506718&#2506718
http://discussions.apple.com/thread.jspa?messageID=729885&#729885

Right, I don't see your model in there. As a test, try the generic "HP Deskjet 900 Series Gimp-Print" driver already installed on your Mac.

What protocol did you use, and if required (IPP or LPD), what queue name did you enter in Printer Setup?

And queue name?
Or what model print server - I'll look it up for you.

For this print server, the queue name is either P1, or, something like PS-xxxx-P1. Try P1 first. If it doesn't work, then use a browser and connect to the print server with:
http://[IP addresshere]
and look in the settings for the "port name" or "printer name." which will be in the form PS-xxx-P1. Then use that for queue name.

Let me know if you get it to work.

Does your 5440 work when connected to the computer with a cable? I ask because I just bought the same printer model and it flat doesn't work (spews endless sheets of blank paper or garbles text or simply Stops Jobs) with either of two Macs running OSX 10.4.6.