Boning around the rim is a good idea to make the bag stiff enough to hold itself open, but boning in the corners would make it impossible to fold the bag up for storage. You can add boning only in the green lines.
A heavy canvas is strong enough to hold the bag open, especially since the material is doubled in the corner seams. You actually don't need much strength to hold the bag upright because its own weight is negligible.
For the boning itself you should use plastic boning. Since the bag is supposed to carry potentially wet items into a potentially wet environment (the pool), you should avoid metal boning because it could rust in time.
You need to cut seperate strips of boning for each edge around the rim. After cutting the right length, clip or file the corners of the boning to avoid damage to the fabric. If you think the boning is not strong enough to make the bag as stable as you'd like, simply add another strip of boning to the same edge.