If it's the stuff I'm thinking of, it had a pretty thin wear layer on top of a high-density fiberboard (HDF) core. It looked pretty, but the 'wooden' core isn't very stable to water penetration. Also, if I recall correctly, the board lengths were all identical - generally a giveaway that it's not 'solid' hardwood. When I examined it in the store, some of their display pieces wouldn't click together, so I'd expect some installation frustration as well.

You need to do a little more archeological digging still. You need to know the whole construction of the floor before you can decide how best to install something new. I'd take up that plywood piece in the kitchen and see what's under it. It's higher than the surrounding floor, so needs to go. You should also be able to see from there what subfloor is under the original kitchen hardwood. It's too bad the space under your entryway is finished, but I'd still do some exploring from below to see what's under there, and to figure out the fix to stop it from squeaking. Is it drywall below or a drop ceiling? I know it probably seems like a pain to tear out drywall, but if you cut out a clean patch, like 2' x the width of the joists, it'll be relatively simple to repair. Just make sure to cut it down the center of the joists so that there's room to screw the patch piece to it later. Cut the hole under the area that squeaks the most, if you can. Have someone walk on and watch the floor from below to see what's moving. It might be as simple as pounding in a couple shims from below, or face-nailing a loose board from above. You want to find out what's underneath the hardwood. It might be laid directly on the floor joists, or there might be 10" wide boards that the hardwood is nailed to. Another thought is to take out a board in the hallway, since that's not original and you want to replace it. See what the subfloor is there. I am a bit puzzled at why, when you've stripped the kitchen down to the original hardwood, it's still higher than the hallway floor. Is your current surface perhaps not the original kitchen hardwood? Peeling back the plywood area will help determine that. BTW, I haven't heard of that stop squeaking product you linked to, so can't offer any personal opinion. And don't pour self-leveling compound over top of hardwood. It needs to go over plywood.

@Beth Sorry, you did say you haven't used it. I read the blog but will definitely read reviews as well. We do need the engineered wood to be on site for a while especially since as we live in a very humid climate. I believe the island wood is maple with a pecan stain. It's just a very neutral mid tone brown stain. I know it's not cherry or oak. Unfortunately the bedroom floor must be ordered asap due to flooding/mold issue. I was hoping to choose something that I could also use in the kitchen when budget permits, as I understand floors should be the same throughout the house. At this point I'm unable to visit stone yards yet. When I do, I will definitely bring the front face of both cabinet drawer finishes with me. So that's why I was putting the cart before the horse, thinking about countertops. The reason I'm getting ahead of myself is because I am trying to create a general direction in my mind. Maybe I should just choose a bedroom floor based on what I would use in my kitchen, since bedrooms and bathrooms finishes aren't as important. Or I shouldn't worry so much about using the exact same wood. I don't know but I appreciate every response you have given me.

Go for it. I would throw down some 3mm or 6mm cork underlay down (it floats just like the hardwood) and then float the hardwood. Use a high-quality joint glue on the hardwood and you should be fine. Besides...if it gives up the ghost in 10 years, would you be upset?