How about the collection of intervals of the form $a \times [b, b+1)$ where $a$ is any real number and $b$ is any integer? These are half-open vertical intervals that are disjoint and the plane is the union of all intervals of this form.
(Of course, you could turn this around to get intervals of the form $[a,a+1) \times b$ where $a$ is an integer and $b$ is a real number, or any of several other such configurations.)
I may have misunderstood the question.