No, Hegel did not consider private property a natural right in the sense in which Locke and others did. The reason is that for Hegel these things are not self-evident truths that appear to our perceptive faculties.
For Hegel, rights occur in a social framework as a judgment of the society that this attaches to a person. To put it another way, rights are for Hegel mediated social objects. At the same time, as a I interpret Hegel (and there are those such as Rorty who differ), these rights are not merely social -- they are actually there in the world. But by world, he means the way that we as rational animals perceive what is out there (Hegel does not believe in immediate sense perception).
To put it another way, think about what a right is. It is either the license for one rational animal to do something without the interference of another rational animal or the requirement that a rational animal cooperate with the task that another rational animal (or group of rational animals) sets forth. This is abstract because it cannot make itself real. As an idea until implemented by rational beings, it does not mean anything without their activity. Moreover, it does not occur merely in nature -- its actualization occurs only in reason.
To give an example to make this make better sense, if you have a right to life, then that imposes on others a duty not to kill you. But It actually does not prevent murder. It merely makes it so that when that right is violated, we judge what has been done to be wrong. Returning to your case of private property, private property is the right to have something for one's own use without others being able to impose on you what you can do with or impede you. But they can impede you unless they actualize your right by granting it to you.
For Hegel, this will have to be just one moment in Absolute Spirit, because the arrangement is already flawed insofar as the maintenance of something individual depends on something social. The solution will be to move to a concept of property that includes society's "rights" to the property and the individual's "rights" to the property into an integrated concept of social right.
- Hegel, G.W.F. Elements of the Philosophy of Right. (1820, recent translation by Nisbet)
- Knowles, Dudley. Routledge Philosophy GuideBook to Hegel and the Philosophy of Right (2002)
- Beiser, Fredrick. Hegel (2005)
Simpler claims from Hegel can also be hinted at in the Encyclopedia (for which the Philosophy of Right serves as an expanded consideration of "right") and the Phenomenology of Spirit