Boeing did largely design build the orbiter, which is the reusable spacecraft that's commonly referred to as the space shuttle, although it was only a part of the entire space shuttle program. Both disasters though were not the fault of the orbiter but caused by failures of the boosters and tank, neither of which were built by Boeing, but these projects are supposed to be designed holistically and so I'd say all the companies involved in that project share responsibility for the shortcomings of the design.

In the case of the Challenger accident, the actual orbiter wasn't the problem. The seals on the solid rocket boosters were. That said, I don't know who was responsible for their design/manufacture.

They love to claim they did as part of their legacy.