Eprint and AirPrint are different ways to print. Eprint is an HP feature which relies on the printer being registered through HP Connect. The user assigns an email address and can share that email address with anyone. There are different levels of security but in essence to print a document a user sends the document as an attachment to the address. HP then forwards the document to the printer.This is exactly what I was thinking, that the system is confused by having the printer both connected to a couple of computers with a wired setup and then also having it supposedly wirelessly connected to another. I don't think that's how it is supposed to work, it should probably be "all or nothing." That, and also it probably would be helpful if all the computers and the printer were on the same wavelength as far as versions of the OS and iOS as well......
The ePrint thing is somewhat different than the wireless capability offered by HP for iOS devices and for computers..... If I recall correctly that is called AirPrint and it is what I use with my computers and my printer. No need to plug anything into an ethernet cable; everything is done wirelessly through my WiFi connection. It is easy to set up and so far for me at least, it has worked flawlessly over the two or so years that I've been using it, even when changing/replacing/updating computers. Works with my computers and with my iOS devices (iPhones and iPads).
That is different from AirPrint which is a direct connection to the printer.
google cloud print is also an option to share printers.
There are lots of ways to share a printer, USB network sharing is just possibly the least efficient one to chose unfortunately.