on the other hand - you already stuck your neck out and gave your real name. So what now do you have to lose?
If you can set this up without any chance of your boss knowing, and that includes any chance this IS your boss (in which case he now knows, because you used your real name) then at least check it out. But I would be extra wary, because this level of secrecy from them, is a worry. What sort of business is it? You don't have to give details, just think about it for yourself: is it the kind of business where they need a public profile of sorts? Or is it possible they are already getting as much business as they can handle (and more) via word of mouth, and are hiring more staff to cope with it? How long have they been in business? Are they registered with the relevant authorities? (if you have the same sort of business registration we do). For example, here in Australia, I could call the ACCC and find out if they are currently registered and even if there have been any complaints made about them. I could also call Department of Industrial Relations and check the registration and reputation with them. You are entitled to contact a prospective employer and interview them beforehand - in my involvement in staffing panels, we often find some applicants who contact the panel convenor and ask for an information package on the school (read 'company' as equivalent) and even a few who have made appointments to come in and look around, even before we've even discussed whether to give them an interview or not. The fact of an applicant coming in for a look-see before the panel has met to discuss the applications, has to be divulged to the panel members and recorded. So it is acceptable, it can happen.
A prospective employer who is not assisting you to find out more, would have to have a very good reason for this. For example, easy child 2/difficult child 2 answering an ad for cast for a film. It was not at all unusual for prospective cast to be told, "It's a short film," or "It's a feature film, big budget." But not much more. They might say, "We are looking for someone to play a 16 yo girl, set in present-day, a girl living on the streets as a junkie. The role will require filming over four days beginning in early March. Location - Sydney." The amount it pays should also be divulged. No other details need be given. Later on if she gets the role she might find out just how big budget (or not) the film might be, who the director is, who any other cast might be. But these are things often kept hush-hush in acting, because they don't want the word to get out, someone else might steal a march on the film, or reporters might get a whisper of it and cause problems by crowding the set (especially if filming is happening in an other wise public place). So thee can be reasons for secrecy, especially if they are concerned about industrial espionage, or some other information poaching.
As I said - there should be good reasons.
But a prospective employer who is trying to hire the best person for the job, for a company trying to compete in the business world, should be putting the word out. Unless the need to hire someone needs to be secret for business reasons (such as a legal accountant to wind up the affairs of a company about to go into liquidation, if they can't mange to trade out) any company will surely want the word spread openly.
If your boss comes back to you and says, "I put a fake job ad out there just to test your loyalty. You failed. Get packing," you can always reply with, "I was not enquiring about myself, I have a friend I studied with, she needs a job, I was making enquiries on her behalf. But she doesn't want her own name used... seems she has a boss who is afraid of losing her, but who won't give her breathing room to do her job." It would be interesting to see how your boss would respond!
Marg