What about this?
eq = {2 x >= y, 2 z >= y, 2 x + y <= 4, 2 z + y <= 4}
FullForm@eq
(*List[GreaterEqual[Times[2,x],y],GreaterEqual[Times[2,z],y],LessEqual[Plus[Times[2,x],y],4],LessEqual[Plus[y,Times[2,z]],4]]*)
MapAt[(# /. {LessEqual :> Equal, GreaterEqual :> Equal}) &, eq, {{1}, {3}}]
(*{2 x == y, 2 z >= y, 2 x + y == 4, y + 2 z <= 4}*)
