Why can Bobbi and Hunter no longer work for S.H.I.E.L.D.?

Remember Mission Impossible? At the end of the orders, there was always the same warning: "If you are captured or killed, the Secretary will disavow all knowledge of your actions." That's what happened to Bobbi and Hunter. They were disavowed.

The trouble with running Black Ops is that you're on the ragged edge of legality. They weren't supposed to be where they were, doing what they were doing. Publicly, S.H.I.E.L.D. doesn't exist anymore, and certainly not as a government entity, so as far as the law was concerned, they were working without any sort of authorization or authority. What they did was successful and saved lives, but at the cost of publicly murdering at least two high-ranking Russian government officials in full view of witnesses.

Coulson's choices at that point were to a) admit S.H.I.E.L.D.'s involvement, which would lead to Bobbi and Hunter's execution and spark an international incident, b) arrange for their escape and relocation and deal with the fallout from the Russian government, but keeping hold of his agents (his choice), or c) disavow all knowledge of them and let their story of "former S.H.I.E.L.D. agents (emphasis on former) in the wrong place at the wrong time acting in self-defense" stand as the truth, and in return, they disappear (their choice).

In truth, the Russians have very little doubt that the story is bunk, but they really can't prove anything else, and they want to avoid an incident as much as the US does. Plus, whatever they have to outwardly portray, they really are grateful that their cabinet minister was saved. So, they agree to drop the charges and release Bobbi and Hunter on the condition that everyone agrees to stick to the official story of them being ex-agents. Which means that Bobbi and Hunter have to leave in order to maintain that image.

At this point in the show, S.H.I.E.L.D. has once again gained an element of respectability after its fall from grace in the earlier series. The President has indicated that they will be officially authorised and will presumably receive government funding. He even puts Coulson in charge of the US Army unit that's been tasked with dealing with the Inhuman threat.

In return, they have to accept even a 'Black Ops' unit is expected to have some kind of oversight. That apparently includes not having these two professional assassins working for them if the US President gives his word to the Russian President that they won't be.