Clever idea ... I am always amazed at the many things one can use to make a slingshot. I do not think those eye bolts are at all likely to come out as long as they are screwed well into the handle. They are actually eye screws, not eye bolts; they are designed to screw into wood. You could screw a couple into a ceiling joist and suspend a basket chair or porch swing from them; and in that case, all the force is along the axis of the screw. In your case, the force is perpendicular (roughly) to the axis of the screw, so there is little or no force trying the pull the eye bolts out of the handle. Where I would have some concerns is that the handle might split from the sideways force on the eye screws. And if the handle splits, you could be injured by the eye screws or pieces of wood. But that looks like a decent chunk of wood. Just examine everything carefully before each shooting session. And by all means, wear eye protection whenever you are shooting any slingshot.
Cheers ...... Charles