I would pick either Boost or MIT for my own code and the overall license of my package, selecting one vs. the other using the one that represents the largest chunk of third-party code I include.
For instance, if the embedded Boost-licensed code is 5000 lines and the MIT-licensed is 1000 lines, I would go for Boost.
My rationale is a matter of courtesy (and ethics): when I embed significant chunks of third-party code in my own projects, I tend to reuse the same license as these projects. If anything, it would allow the original author of the embedded code to eventually reuse any of my improvements in their own code, should they wish to do so.
But licensing a combination of MIT and Boostv1 code with either the MIT or Boostv1 license is alright?
Yes. I would be licensing my overall package for instance as Boost, including all my new additions. Existing code under the Boost or MIT license would stay under their original licenses and carry with them their original copyright and license texts and notices.
In your hypothetical situation, aren't you violating the MIT license by requiring that "The [MIT] copyright notice and this permission notice shall be included in all copies or substantial portions of the Software." Your hypothetical situation would definitely be a copyright violation if we were talking about the GPL v3 rather than the MIT license.
No. I would still keep the bit originally under MIT or Boost under their original license. Only the overall license for my package and the license for my additions would carry the license I picked.