Loctite PL300 Foamboard is good, but Loctite PL Premium also says it glues XPS (Pink) foam - Product page, see tech data sheet – I have used both in the past few days (Foamboard to glue pieces to each other, and PL to glue foam to wood) – but I didn't remove any laminate. Both glues seem OK, but "mechanical" fasteners are often called for. In other words, while PL glue is stronger than wood fibers themselves, I don't think you're gluing foam into a singular unit, and most instructions are about attaching foam to something, not making a crate.
Also, I know from experience that animals don't mind ripping the pink foam to pieces (mice, rats, racoons, groundhogs) if they decide that's important to them.
Also, that stuff is EXPENSIVE compared to a sheet of even high-quality OSB, which you could nail and glue into something really strong – but maybe you're trying to make something simple. Here the PL glue would come in more handy because it glues lots of things together, you could even glue the box to concrete. Costwise, thinner insulation and OSB together might be stronger, and not blow away. I guess you're trying to add some insulation, but you could just get old clothes or blankets – I'm sure feral cats aren't picky.