Because of her mother's job, May grew up in a world of grey and her worldview certainly didn't change when she joined S.H.I.E.L.D. Coulson was a rare white knight in that world. Worrying about the big picture, about morality, had always been his job. She could trust him to make the hard choices, knowing that he was a good man, so she could do her job and be a perfect soldier. That faith was doubly important after Bahrain.
May is so used to seeing him as a rock - nothing fazed him, not alien invasions or even Tony Stark - that she's shocked when Phil starts to lose control. And it isn't a physical reaction to the procedure; it's a mental reaction to the truth. Then S.H.I.E.L.D. crumples and they find out just how deeply HYDRA was embedded in their midst. It turns out Coulson's the same as she is; it's just his faith was in Fury and in S.H.I.E.L.D. He's ultimately still a good man, but he's not perfect. He's not her rock anymore.
So she speaks up, questions his orders for the first time. When Skye, who also saw Coulson as her rock, sees him faltering, she comes to May. She might not always agree with May, but she trusts her not to lose it. And yes, May is shocked by what is happening, but she is not shattered like after Bahrain. She doesn't need a rock anymore. So when she gets fed up with how Coulson's treating her, she leaves.
And, yes, she comes back, because he deserves answers and because HYDRA needs to be stopped. She can make the hard choices now - or at least realized she can. She doesn't need him to tell her what to do. She doesn't need him to be her rock. She even seeks out a heartbroken Skye because Skye needs someone to be there for her. May knows she is at least partially Skye's rock now.
And, yes, she'll probably stay and follow Coulson in the fight against HYDRA because she decided that this is a fight important enough to put above her wants or needs. Because she does believe in him, especially since he's got his head straightened out now that the soldier in him has had his faith in Fury and in S.H.I. .'s purpose restored. She'll follow Coulson - and question him if need be - because she's not just a perfect soldier, she's a good woman.