It appears that disruption was because of a distributed denial of service attack (DDoS) using the same Mirai botnet vulnerability that took down huge swaths of the internet a few weeks ago. "I am a tech specialist who currently has access to a slightly weakened version of the Mirai botnet," a 4chan user who goes by "Sparky" wrote on the site's /pol board. He also said he took out all of Clinton's phone lines in Nevada and asked other 4chan users for more targets.
NextGen's Suzanne Henkels confirmed the issues, telling The Verge that "[our tool] was slower in the morning, and then went down for hours at a time." The attack was focused not on NextGen specifically but the Utah-based call center provider TCN -- which is also used by groups like MoveOn and Our Revolution for automated calling, though it's not clear if those groups experienced the same disruption as NextGen.
The call center wasn't the only victim of a Mirai botnet DDoS attack -- both Clinton and Trump's sites were hit with attacks and experienced outages this morning.