Thanks Buzzsaw.......along with the advice, the store sold me a 5 gallon pail of the stuff ($28), I am definitely a sucker for a bargain....If you already had the glue would you be more inclined to use it? I appreciate your honest answer.
One of my main concerns is where I am going to apply the stones....along a 1/2 wall on the second floor over the stairway, and on a heavy beam above the kitchen doorway. I would hate it if one of the stones came loose and bonked somebody on the head ( I would REALLY hate it). Do you see this happening? Thanks again for your reply and know-how!!!

Is this a stacked ledgestone that you are doing? You might be able to get away with the adhesive but I think you will likely end up with some gaps that the mortar would cover (if used) but will be left exposed if you are going with the adhesive. The adhesive I typically use for various projects is PL Premium tubes and while it would work to hold it in place, I think you will likely experience some "slide" factor and the stones will not stay in place where you put them but will have a tendency to slide down (you can partially remedy this by putting in a blocker (2x4) for your starter course that you can remove later). This will likely only work for ledgestone or other tight fit looks. If you are doing anything that has grout lines, the adhesive will leave those grout lines exposed and it will look really "faux".
I would be concerned that if the drywall got wet the paper could peel away or might just do so with the weight of the stones (though fake they are still somewhat heavy). You may very well end up with falling stones....
I would suggest renting a scaffold and going the recommended method of applying felt paper and the mesh and mortar (you can likely skip the scratch coat and do one coat while buttering the stones), ensuring you hit studs while nailing the mesh in (unless you're nailing onto plywood).

If I was to use something 'new' or less proven product on an overhead project, I'd be pretty leary to say the least. At a minimum, I'd get a couple rolls of 2 inch wide tape to tape the stones together and/or to the ceiling or beam. THat may give you a little extra support while the adhesive sets up. Once its cured, I'd personally wiggle every single stone to verify it was solid.

Demarara brings up a good question. Are you going to grout the joints? If so, I wouldn't bother with your 'bargain' adhesive at all since you're going to have to deal with essentially the same type of mixing, mess, and work of using mortar anyway.

"The folks at my local "Restore" were very certain that for an inside application I could use a heavy construction adhesive- very similar to liquid nails-to install cultured stone."

Are they wiling to replace everything (materials and labor) if the installation fails?
IN WRITING?

Considering that the gap-spanning ability of construction adhesive is rather limited, I would not risk using it to install stonework.

There is a right way to do the job. It's the "right way" because people have been doing it for many years, and it has proven that it works well.

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I just reread some of the other posts on this thread, and saw a reference to "drywall paper." Is that for real??? Are you actually planning to have this stone hanging from the paper facings of drywall??? Please tell my you're not seriously thinking of supporting stone with paper.

Instead of all the speculation, call Eldorado's help line at 1.800.925.1491 and get the correct answer. Even if it means returning that bargain pail of adhesive.

OK, OK, OK! I'm hearing a lot of good advice, even if it's not what I wanted to hear! Thank you all. Of course the store won't put anything in writing as far as guarantees, and I doubt seriously that they'll even take it back. Sounds like I have to go back to the 'tried and true way", and I know it's that for a reason.
Anybody have suggestions as to what to do with a 5-Gallon pail of sticky, construction adhesive??? LOL? Thanks again, all, for saving my project, and -who knows?- , maybe even a life! I appreciate your help.