What G forces does a pilot perceive in a "1 G" barrel roll?
You can't do a barrel roll at 1G.  It's a loop pulled out into a coil, and you have to pull pitch to get it started so you are going to be 1.5 to 2 starting out, declining to 1G to .5 at the top, and pulling 1.5 to 2 on the pullout.

So the sensation at the start is like a hard climbing turn (which is what you're doing), pressed into your seat with about the force of a steep turn at the start, declining to normal gravity or bit less (giving a bit of that giddy becoming weightless sensation, but not coming out of your seat) as you ease off the pitch coming around the top, and about the same pressed into the seat force as at the start, or a little more, on the pullout. 

A good barrel roll should be no more than 2Gs peak, but how much is a function of pilot skill and of the airplane's ability to keep up the energy through the first half.  A jet with oodles of power to keep the speed up in the steep rolling climb can get away with a very gentle pull and will see only about 1.5 or less on the pull up and pull out.  

But it's always going to be more than 1 starting out and finishing off and when Tex Johnston said in an interview it was a "1G maneuver", he really meant "not much more than" as in less than 2 (and no worse than a steep turn, unless you screw it up and fall out).  Watch the film of the barrel roll and you see a firm pull up at the start with a fairly steep nose down attitude on the exit, which would have required a pull of at least 1.5 G in returning to level flight without over speeding.