Wooden chairs that consist mostly of lathe turned wood are very difficult to clamp, especially where most of it is tapered as well. Not only will the clamps slide off, but also often damage the surface. Forget the clamps; there is a much better way.
Find an inner tube. Most tire repair shops will have one lying around that you can have for a smile and little money. If you cant find an old auto inner tube, a tube for a wheelbarrow or utility trailer tire isn't too expensive; also nice to work with.
Okay, so you found a tube. Cut it square across in two places, so you have a piece of inner tube 6 or 8 inches long or whatever you like. Start cutting from the end, around and around till you got a long rubber band, several feet long and ½ to ¾ inch wide. Thicker rubber is hard to work with if it's too wide. Make this length at your own discretion. The main thing to watch as you cut around, is try to keep the width fairly even; a clean cut, no deep nicks on the sides, as this can start it tearing.
If the wood is broken square across this will not work; forget about it!
Most chair legs and spindles seem to split at an angle following the grain of the wood, so if you have even a fairly sharp angle, it might still be fixable. Make sure the split fits together nicely and then apply a bit of good quality wood glue to both breaks; not too much, just enough to nicely coat the break, but if you get too much on, it'll just squeeze out; just more to scrape off after.
Now squeeze the pieces together and wipe off the glue that oozes out. Hold it there a minute as it starts to bond when you do that, and then start wrapping the rubber band around it, pulling it tightly as you wrap. Keep wrapping back and forth, several layer till you run out of rubber band. The pressure exerted on the glue joint is so horrendous that you won't even find the break when this is done. Keep it warm and let it set for several hours. Go for it girls. Nimble fingers work the best. Jonsig.