Does the sporophyte generation make egg and sperm cells?

Answer 1

No ! As the name implies the sporophyte generation only produces haploid spores.

On germination these spores produce the gamtophyte;

and as the name implies once again the gametophyte produces the gamete in the form of eggs and sperms.

Both are haploid , as they fuse; once again the diploid condition ( Sporophyte) is restored.

This is called alternation of generations.
http://img.search.com/thumb/8/86/Sporic_meiosis.png/400px-Sporic_meiosis.png

Answer 2

The sporophyte generation gets its name from the fact that it reproduces by making spores.

Sporo = spores
phyte = plant

The generation that makes eggs and sperms is the gametophyte. Eggs and sperms are called gametes.

Answer 3

i do agree with ecolink