I'm not a professional, but I've paid to have flagstone laid twice - at our old house & new house - 2 different contractors.
When was your house constructed?
The reason I'm asking is if it's newer, the dirt around your foundation could still be settling. The longer you wait, the more secure the job.
The concrete method may be more expensive, but you get what you pay for... The blocks could sink & shift unevenly, depending on the dirt under them, & you'll be wanting to re-do your steps sooner. By a back door it may not be such a big issue if you don't use the door as much.
Ask for references & addresses of jobs they did awhile ago, not just recently. If you see the homeowners outside, go up & confirm that company really did the job as they claim.
Ask about the warranty & get it in writing.
Ask how they handle sinking stones, etc & for how long.
If a stone shatters due to bizarre freeze/thaw issues, will they replace it?
Make sure they leave a few left-over flagstones with you in case a stone does shatter & the contractor disappears or won't return your calls.
(Or someone in your family spreads the wrong ice melting salt on your stones...)

artdeco
Thanks for the input. Our home is older (35 years) so the dirt has already settled. The one contractor who recommended masonry block instead of concrete doesn't lay concrete.

The longest warranty I have been able to find a contractor offering is 5 years. Everyone else appears to be at around 2 years. Not very long if you ask me.

A lot of this can be based on an individual contractor's comfort zone. There is nothing wrong with blocks on a footing or poured concrete. Ideally, these should be on a footing below the frost line, although more often than not, they will be on a 6" steel reinforced slab footing just below the surface.

Aesthetically, why not go with mortared stone or brick on a footing?

I installed my own dry-laid 1.5" PA bluestone patio last summer. I ran a vibraplate tamper on the bare soil and then again after the 4" gravel (did not do it half way, like you asked). I did not use landscape fabric or edging. I used about a 4" layer of masonry sand.

I don't have a recent pic handy but I posted a midway pic from the summer below.

This was my first patio, but I have a stone mason friend who's done tons of them and he said it's really solid and that my foundation is probably more burly than what they normally do. It's not completely finished, as I just have topped off sand in the joints - trying to get sedum/thyme established this spring and may also fill with stone dust.

As far as durability, I first tested it with my car and a friends minivan - no problems. I then drove a pickup with a 600 lb tree in the back over it - again, didn't budge. However, I got a little too confident when I tried to get a delivery of a cord of firewood driven over it. Had a few nice big stones shatter, but this was mainly due to a drainage problem I encountered on the car path. I get a lot of water runoff from my front yard into this sloped area and have lost a lot of sand during rain storms, and the stones that cracked had pockets of missing sand. Looking to fix the drainage this spring, but for now I have to lay tarp over the driveway section whenever we expect heavy rains.

Other than that, it's rock solid, especially when the ground freezes. Once I fix the drainage, settle on a solution for the joints, and get my plants going, I think this will be good to go. And it already has a weathered look like it's been around for years. Message me if you need any more info and maybe I can find some more pics. Good luck, it will be worth it when it's finished!

PS - I should also note that I was really picky with getting the right pitch away from the house. I used to get tons of water in my basement but ever since I put the patio in, it's been practically bone-dry.

Here is a link that might be useful: {{gwi:53255}}