If I purchase seeds of a crossed plant, do I own the rights to it, Can I register the plant?
The seeds of a crossed plant will not come true. However, if you got something unusual, you might own the rights to it. I'm not sure what you mean by "register the plant".
The answers is no. I'm assuming you are talking about a F1 hybrid that someone already has the right to.
Read the first and second paragraph: http://en.wikipedia.org/wiki/Plant_breeders'_rights
"...rights granted to the breeder of a new variety of plant.."
"...a variety must be new, distinct, uniform and stable."
"A variety is stable if the plant characteristics are genetically fixed and therefore remain the same from generation to generation, or after a cycle of reproduction in the case of hybrid varieties. "
Depends on what is meant by "crossed plant".
Also depends on whether the plant is an annual and produced only by seeds or a perennial (or some annuals) that can be produced by seeds or vegetatively.
And may depend partly on whether the seedlings are different enough from the parent to not be considered "essentially derived varieties".
Can you please provide more detailed information about the crossed plant.
I agree with Joe.