Does it have to be one of those HP printers? No. It can be a Brother, or an Epson, or others.
First, did you read the page at Apple, regarding printing from your iPhone?
Second, after reading that page, at the bottom, did you follow the link to AirPrint-Enabled printers that was provided?
Printers not listed in this article are not supported by AirPrint.
Note: Some printers listed in this article will require a firmware update out of the box to become AirPrint-enabed. See the manufacture's documentation or website for details.
The AirPrint-enabled printer must be connected to the same home Wi-Fi network as the iOS device. This is the default configuration in most home Wi-Fi networks.
Now, assuming that you DID both of these things, and you did see the model printers that were being offered by all those different manufacturers... what is your actual question. Are you asking if one of those printers listed there actually works to allow printing from your iPhone?
If you really want to stick with an HP, it can be any one of these...
HP Deskjet 3050A J611
HP Deskjet 3054A J611
HP Deskjet 3070A
HP Envy 100 e-All-In-One D410
HP Envy 110 e-All-In-One D411a
HP Envy 114 e-All-In-One D411a
HP LaserJet P1102w Printer
HP LaserJet P1606dn Printer
HP LaserJet Pro 100 color MFP M175nw
HP LaserJet Pro 300 Color MFP M375nw
HP LaserJet Pro 400 Color M451nw-M451dn-M451dw
HP LaserJet Pro 400 Color MFP M475dn-M475dw
HP LaserJet Pro CM1415fn Color MFP
HP LaserJet Pro CM1415fnw Color MFP
HP LaserJet Pro CP1025nw Color Printer
HP LaserJet Pro CP1525nw Color Printer
HP LaserJet Pro M1212nf MFP
HP LaserJet Pro M1213nf MFP
HP LaserJet Pro M1214nf MFP
HP LaserJet Pro M1216nf MFP
HP LaserJet Pro M1217nfw MFP
HP LaserJet Pro M1536dnf MFP
HP Officejet 4620 e-All-In-One
HP Officejet 6100 e-Printer H611a
HP Officejet 6500A e-All-In-One E710a
HP Officejet 6500A Plus e-All-In-One E710n
HP Officejet 6600 e-All-In-One H711a
HP Officejet 6700 e-All-In-One H711n
HP Officejet 7500A Wide Format e-All-In-One E910a
HP Officejet Pro 8100 e-Printer N811a
HP Officejet Pro 8500A e-All-In-One A910a
HP Officejet Pro 8500A Plus e-All-In-One A910g
HP Officejet Pro 8500A Premium e-All-In-One A910g
HP Officejet Pro 8600A e-All-in-One Printer N911a
HP Officejet Pro 8600A Plus e-All-in-One N911g
HP Officejet Pro 8600A Premium e-All-in-One N911g
HP Photosmart 5510 e-AiO B111a
HP Photosmart 5514 e-AiO B111h
HP Photosmart 6510 e-AiO B211a
HP Photosmart 7510 e-AiO C311a
HP Photosmart e-All-in-One D110a
HP Photosmart eStation C510
HP Photosmart Plus e-All-In-One B210a
HP Photosmart Premium e-All-In-One C310
HP Photosmart Premium Fax e-All-In-One C410a
HP Photosmart Wireless e-All-In-One B110—Europe and Asia-Pacific
HP TopShot LaserJet Pro M275