This claim is false.
While the restriction of women being able to join the military was lifted as of January 2016, females are still not required to register with the Selective Service. According to the Selective Service System, "Defense Secretary Ash Carter announced on December 3, 2015, the Department of Defense will lift all gender-based restrictions on military service starting January... As of January 2016, there has been no decision to require females to register with Selective Service, or be subject to a future military draft."
Currently, the Selective Service is not active and there is not a mandatory draft sign up. According to The Washington Post, "On Jan. 27, 1973, with the Paris Peace Accords signed and U.S. involvement in Vietnam over, Defense Secretary Melvin R. Laird announced the end of the military draft, after 25 uninterrupted years of conscription."
Additionally, U.S. officials have stated that as of October 10th, the day the article was written, there is no sign of the draft being reinstated. According to AP News, "Defense officials confirm the armed forces have made no such recommendation to Congress or the president, which are the entities empowered to authorize a draft."