Cook rhubarb chunks in the small saucepan with sugar and water until the rhubarb is fork soft, about 20 minutes. Mash with a fork and allow to cool completely. If the rhubarb is stringy, you can puree it with an immersion blender or in a jar blender.
In the plastic bowl, whisk together the Greek yogurt, half & half, granulated sugar, and vanilla extract. Stir in the rhubarb puree until it’s thoroughly mixed in. Cover and chill in the refrigerator at least 4 hours to overnight.
About 30 minutes before churning, put the yogurt mixture in the freezer to get it really cold. Churn in your ice cream freezer according to the manufacturer’s directions. Add the strawberry chunks a little at a time during the last 2 minutes of churning. Scoop the frozen yogurt into the plastic freezer container, press a piece of plastic wrap over the yogurt, and put the lid on. Freeze until firm, at least 2 hours, before serving.