I have been using the pocket hole system for years, and they do make the fasteners for 2 x stock which will give a much stronger joint than just using regular screws because they are:
1) Stronger screws
2) Smaller heads
3) Predrilled holes that will not cause wood splitting.
If you just use a regular wood screw or a sheetrock screw in a piece of 2"x2" lumber, you will only be able to safely use 1 screw per joint, otherwise you WILL split the wood rendering the joint very unstable, especially for something outside.
This system has been updated and is using different fasteners from what it used 4 or more years ago, and is suitable and highly desireable for outdoor use now.
I was a custom home builder for years and have quite of bit of experience with it and would not hesitate to use it for this purpose. If you want a little more insurance with it, get a bottle of gorilla glue at Lowes and moisten the ends of both boards being fastened first, put a dab of gorilla glue on and then screw. The gorillla glue is a polyurethane glue that is weather proof and will hold up for years.
Using this method will provide a much stronger joint that will be far superior to wood screws penetrating through the ends of a joint that will be exposed to the weather. Even painted, the exposed screw heads will allow moisture to absorb into the end grain which will cause even more wood splitting and joint failure.
Pat, she indicated that she was using 2x2 for the framing. If it was 2x4, then 2 wood screws with your method would be quite different and an acceptable approach. But, when you split that in half to a 2x2, wood splitting will be an issue if you try to use more than one wood screw per joint, and that is not enough to give you a strong joint. Your pieces will continue to turn on you.