If you take the brake shoes out, it will not be like a free wheel. it will act the same as it does with a coaster brake, it just won't stop.
You'll still be able to push the pedals backwards, and they will stop, but the rear wheel will not stop spinning....you will have no brakes at all.
To me, from what you described, it sounds like you don't have the bearings adjusted right.
There are 2 nuts that screw onto the axle (on the side with the sprocket). The innner nut is what the bearings ride against on the inside of the hub (cone). The outer nut acts like a jam nut. You tighten the inner nut to get the bearings adjusted right (not too tight, not too loose) and then you tighten the outer nut so it jams against the inner nut. You want to loosen the inner nut slightly and tighten the outer nut against it so they jam together. You have to do this in such a way so that your bearings are still adjusted right. This will keep the inner nut from coming loose. When/if you slide the axle assembly out of the hub, the brake arm and dust cover should stay connected to the axle. You only have to take off the 2 nuts on the side that has the sprocket to get the axle out.
I think what's happening is that the 2 nuts on your axle are not jammed together, so when you pedal backwards to stop, they want to loosen up.
Also, is your brake arm attached to the frame? If the brake arm is just floating there and not connected to anything, it will want to spin when you try to use the brake. If the brake arm spins when you try to brake, it will cause your axle to rotate backwards, which will loosen the 2 nuts on the sprocket side. Eventually they will loosen up enough for bearings to fall out.