Yes. According to the nLab page for the Strom model structure, this is proven in Section 6.4 of May's Concise Course in Algebraic Topology. 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 paper.
David White
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