Yes. According to [the nLab page for the Strom model structure][1], this is proven in Section 6.4 of May's [Concise Course in Algebraic Topology][2]. The point is that a closed inclusion is a cofibration if and only if it's an NDR pair, and you can use that to check that the pushout corner map is a (trivial) cofibration as required. Also, Donald Yau and I made use of this fact in Example 4.5 of [this][3] paper.


  [1]: https://ncatlab.org/nlab/show/Str%C3%B8m+model+structure#monoidal_structure
  [2]: https://www.maths.ed.ac.uk/~v1ranick/papers/maybook.pdf
  [3]: https://arxiv.org/pdf/1703.05359.pdf