Reactionary Southerners said they were fighting was southern state governments run by African Americans. Does the data support this claim?
It would really be quite difficult to support this statement. To say that blacks dominated state governments at any point of Reconstruction is simply not accurate.
To be sure, blacks had much more of a role in government than they had ever had before. The link below tells us that around 600 blacks were elected to state legislators which, when compared to the situation where blacks could not even vote, was a huge change. These blacks did form coalitions with white Republicans and could be said to have had an impact beyond their numbers. Even so, the fact remains that only one house of one state's legislature ever had a black majority. This was the lower house of South Carolina's legislature in the early 1870s.
White Southerners made this claim as a way to discredit the governments against which they were fighting and to try to use claims on racial solidarity to help them gain support in their attempts to "redeem" the South.