I like that plan, Ralph. Is there plenty of good, solid wood behind the box? I'd use lag screws - as large as possible. And of course you'll want to predrill the wood so it doesn't split.
Good solid wood? I sure hope so, but it's anybody's guess. I'll know when I drive the first screw. For sure I'll pre-drill the holes. If the wood is solid, it will be dry and hard. I found flat head wood screws that seem to be the perfect size, but since they have slotted heads, pre-drilling is a must. If I have any trouble, I'll go with lag screws instead.
You might want to look at some deck/construction screws. There are some types that have special drive sockets (and the box comes with an adapter). Anyway, they can be tightened more than those with standard philips or even slotted heads. They are kind of pricey, though.