Generally speaking, American citizen must have a clear intent or understanding that their actions will renounce their claims to citizenship before their rights may be taken away. The case of Afroyim v. Rusk, 387 U.S. 253 (1967), which involved an American citizen voting in an Israeli election, solidified this concept. Because those born in the United States have a constitutional right to citizenship, any attempt to unilaterally deprive them of that right without a court proceeding would be a violation of the Due Process clause. Afroyim v. Rusk stands for the principle that Congress cannot divest someone of citizenship without a showing of intentional renunciation on behalf of the citizen.
Without the intentional (and willing) revocation of citizenship, a citizen retains their rights. The Nishikawa v. Dulles, 356 U.S. 129 (1958) case illustrates where the choice was intentional, but not willing. Nishikawa was a dual Japanese-American citizen before the outbreak of World War 2. He was conscripted into the Japanese military. Avoiding conscription would bring a penalty of 3 years imprisonment. The court held that the government failed to prove that Nishikawa willingly revoked his citizenship, even if the service was intentional. This point is particularly salient when considering the case of Yaser Esam Hamdi, an American citizen who was captured fighting in Afghanistan and was eventually released to Saudi Arabia after he renounced his citizenship.
I do not believe that this is a viable solution for addressing the issue of drone strikes on American citizens in these war zones. First, the divesture of rights would violate the Due Process clause by unilaterally stripping constitutional rights without a court proceeding. Second, even in situations with a court proceeding, the citizen could argue that they did not intend to renounce their citizenship, possibly claiming that their actions were done with the knowledge that they might face the legal consequences under U.S. law.