> Is it possible to get my sequence from one of the commercial groups doing this, and then "look it up" in a genome reference to verify my brown eyes, for example?

Yes and no.

Brown eyes aren't controlled by a single gene, so it's not straightforward to look this up from your genome profile.

What genome profiling companies will do is assay a finite number of specific locations on your DNA (SNPs), to get good comparison points with other people, and then predict what colour your eyes are based on which SNPs are significant for eye colour.

It's not guaranteed to be right, then - nor is it guaranteed that the significant SNPs are linked to relevant genes.