Well, if you have to break a rocker, that's the kind to break. The pictures are kind of dark, but it looks to me like it is half-lappped onto the bottom of the legs, rather than the kind that the legs have a tenon turned on bottom of the legs; with those you have to disassemble the chair to get a new rocker on. Could you get me a better picture of just the breaks, with some light on it?
It looks like cherry; is it?
Gluing it is certainly worth trying. The longer break looks new and clean, it should work fine. But the shorter one just under the leg post looks like it's been broken for a while (unless it's my imagination and the lighting), it may not be as successful. If it has, there will be dust in the wood on the mating surfaces, and probably some wear - which will prevent a perfect glue joint. And to have a glue joint stand up to two people rocking, it must be perfect. The type of glue doesn't matter, all it needs to be is stronger than the wood, which all woodworking glues are. But glues don't have any shear strength, which is why the mating surfaces must be perfect. If the glue is filling a gap, it will fail.
My major professor was German; between the two world wars, he had a job building wooden airframes, biplanes with doped canvas skins. The glue used was hide glue; he said they were required to glue joints together within a half-hour of being freshly scraped, or the mating surfaces had to be scraped again. Joints have to be perfect.
I would glue the long break first; don't try to do them at the same time, more chance for error. As always, before you use any glue, do a test clamping, dry, to be sure how you're going to do it. Use two small C-clamps on the flat sides to make sure they don't drift out of the same plane. (Be warned, when you're gluing two tapers together, the glue initially acts like a lubricant, and they will slide apart.) Use some scraps of formica or plexiglass under the clamp heads to keep them from denting the wood. Then figure out how you're going to clamp the wood top to bottom without denting the wood. A lot of bar clamps come with rubber covers for the clamp heads.
For the front break, I would remove the wooden peg that holds the rocker to the leg. But I'm going to stop now, until you post a better picture of the front break. I didn't set out to write an essay, this already more than people want to read. I'll continue later.
At worst, it's not that big a deal to make a replacement rocker, I can walk you through it.