"\\192.168.2.2\printer1"
That's more than just an IP address. It is a Windows share name (the name of a printer share, in this case).
So 192.168.2.2 is probably not be the IP address of the actual printer, but of a Windows computer that is connected to the printer.
If you haven't connected your Mac to any other HP printers yet, you might have to download and install this HP printer driver package from Apple first:
http://support.apple.com/kb/DL907
Try printing a test page using the printer's control panel. According to the documentation for that printer model, there should be a "Network Address" button on the control panel (either a real button, or more likely a "software button" on the touch screen). Touch it, and then touch "Print". The printer should print out a test page that contains a lot of information about the printer settings. Look below "TCP/IP": the IP address of the actual printer should be listed there. That is the IP address you should be trying to configure to your Mac.
But if the network administrator has been serious about the requirement to use username and password when printing, that IP address might not be reachable from your Mac, only from the "official" print server (the one with IP address 192.168.2.2). Or the printer might not have an IP address at all: then the only way to access it is through the Windows print server.
To connect to the Windows print server, you might try these instructions from Apple:
http://support.apple.com/kb/PH10880
But if the Windows administrator has been serious about security (i.e. you have a pure Active Directory-based Windows network, with legacy NetBIOS browsing disabled), then you'll probably need this procedure (really good instructions here, with pictures!):
http://its.truman.edu/documentation/index.asp?docId=424
(To sum up: Apple has been sneaky and hidden the advanced feature you'll need. The "Advanced" option is omitted from the standard toolbar of the "Add Printer" window, and you'll need to right-click the toolbar and select "Customize Toolbar", then put it back before you can use it.)
In step 7 of that procedure, you should type:
smb://192.168.2.2/printer1
That is the name of the printer share, translated to Unix-compatible form: the share name has been prefixed with the "smb:" protocol identifier, and the backslashes changed to regular forward slashes.
Your Mac should then ask for the username and password, either when you complete the procedure described in the above-mentioned link, or when you print to that printer for the first time.