Simple answer: yes, they were considered citizens. The position held by Unionists and later confirmed in Texas v. White was that secession had no legitimacy, as Texas (and the Southern states by extension) had not declared independence but instead a constitutional remedy for a set of grievances that many felt did not warrant the extraconstitutional/natural right to revolution. The official position of Lincoln and his Cabinet, the SCOTUS under Chase and prior, as well as the majority of Congress, was that no such constitutional solution existed—a position dating back to the ratification process itself (e.g. Madison's assertion in a letter to Hamilton during the NY ratification convention, among other places, that ratification was a permanent measure; this can also be found in an 1833 letter from Madison to D. Webster, complimenting him for his Second Address which upheld the permanency of the Union.)

I can qualify this with better sources if necessary, but the explanation Lincoln gives in the Spielberg film does a very illuminative job of explaining the rationale behind this stance. Essentially, Lincoln rejected Compact Theory in favor of the idea that the power of the Constitution was lastly predicated of the will of the people themselves rather than the states. He held that the states and their governments were not themselves separate from the Union, invalidating the Confederacy's status as an independent nation while upholding the citizenship of all those living within those states which had seceded. It's for this reason that Unionists held that the conflict was a rebellion or a civil war, rather than a war between nations. (There's more to this scene, as it delves into the legal conflict Lincoln himself was aware of in his issuance of the Emancipation Proclamation, but that's not as important in answering your question.)

Edit - Here's another response I gave on a similar topic, which touches on some of the same points I've made here in more detail (specifically concerning the debates during Ratification and the Antebellum period).