Can tile or click-lock wood be installed directly on rigid insulation foam
Just my 2 cents.  

If you are doing a floating engineered floor over the foam, it might work if the foam doesn't flex and expand/shrink very much. You don't have to glue it to the foam as the click-lock mechanism keeps the pranks together.

For tiles, for sure, you can't do it over OSB/Plywood as it will expand/shrink and cause your beautiful new tiles to crack.  You mentioned your floor gets cold thus I imagine you live in an area with fairly good temperature differential. Thus this problem will happen sooner rather than later.

I can't comment on the foam unless I know what type of foam you are considering. But I can say I have never seen any tile installation over foam. When it comes to tile, preparing the substrate is half of the work.  We always tile over concrete, cement boards, or uncoupling membranes. 

In addition, ensure the thinset can bond properly to the foam. Calling the manufacturer would be best.