I would not use 1x2s for the crucial parts of the pen -- the part that lies on the ground, or the corner uprights. You could quite likely get away with GOOD QUALITY 1x2s for the rest of the pen, making sure to orient it the correct direction for each part so that it has maximum strength. However the result will be weaker (a problem if, like, a dog starts jumpin' up and down on it, or it gets blown over, or you bash it into something while moving it) and rot will be more of a problem sooner.
The smaller the mesh (and heavier the gauge of wire) you use, the better it will brace the structure and to some extent compensate for a weaker frame.
Remember though that the extra weakness of a 1x2 frame is not just because of the thinner wood -- it is in large part because your screws or other fasteners are not 'biting into' very much wood in a 1x2. Thus it is much easier for them to split out and things break apart.
Doubling 1x2s to make a de facto 2x2 is possible but probably not a good idea since real 2x2s are so easily available -- the result is not quite as strong, and definitely not as rot resistant as a real 2x2.
You will never regret having some spare 1x2s sitting around -- so personally I would not return them, just go get more 2x2s, unless money is *really really* tight.