Thankfully there are some things working through the legal pipelines to finally answer this question:
No, no one can own the information about the DNA in your body.
The genome is the sequence of nucleotides (A, T, G and C) in every cell in our body that, in particular orders, give the instructions for what a gene will do, where it will be expressed and when.
Although we sequenced "the human" genome, each individual has small differences that make each of us unique. These differences can be a single nucleotide change (for example, a change from an "A" to a "C"), or an insertion or deletion of sequence.
There can be some confusion when discussing "alleles" related to certain genes, specifically with respect to diseases like cancer, that I'll try to clarify here.
For the most part, we all have the same set of genes. Every human has the BRCA1 gene. It is located on chromosome 17:
However, there are small differences in the sequence of this gene between individuals. Each unique change or set of changes to the BRCA1 gene can be described as a different "allele" of that gene. I like to think of a particular gene as ice cream, and alleles as flavors of ice cream. You can have many different flavors of ice cream, but the frozen deliciousness you are eating is still ice cream. Think of it this way (purely made up example):
Vanilla = Normal gene sequence = BRCA1_allele1
Cookie dough = Mutation from A to T at position three = BRCA1_allele2
Chocolate = Mutation from C to A at position seven = BRCA1_allele2
Just because there is a small difference does not mean that there is a new gene, it is just a variant (or allele) of the same gene.
So, why do people want to look at the BRCA1 gene, with respect to breast cancer? Well, most of us have the vanilla allele - our BRCA1 gene works fine and doesn't cause any trouble. But, researchers identified an allele, we'll call it the Mint Chocolate Chip allele, that misbehaves, and is linked to a higher incidence of breast cancer.
What private companies have been doing is trying to justify patenting the sequence of the BRCA1 gene, that they own the right to knowledge about whether you have Vanilla, or Cookie Dough, or Mint Chocolate Chip.
"Precedent indicates that laws of nature or natural substances can't be patented, although significant transformations to natural products can lead to a patentable process or product. The USPTO grants patents to isolated DNA based on the conclusion that they are "distinctly different in character" from the DNA that is present in human cells."
What private companies have been doing is trying to justify patenting the sequence of the BRCA1 gene, that they own the right to knowledge about whether you have Vanilla, or Cookie Dough, or Mint Chocolate Chip.
It looks like now, this will be changed.
Good.