Will this G.SKILL 16GB 2x8GB DDR3 1600 work for an H77 motherboard?

I don't see why not, it doesn't matter the brand of RAM as long as it is a speed that your motherboard supports. DDR3 1600 Mhz is pretty standard now so it should, most motherboard manufacturer web pages give you a RAM compatibility list and it shows what they have tested for the motherboard. Also, make sure that your motherboard can support 8GB per slot, some motherboards only support 4GB per slot. Check those out and you should be fine.

I do not see Intel H77 in the list of Motherboards supported.