It was not a sick out. The article clearly states:

“The sick call rate over the weekend was no different than it has been since June and the pilots actually picked up open flying as well,” Amy Robinson, a spokesperson for SWAPA, told TPG on Monday.

Southwest management doesn't have a legal incentive, but they have a business interest: they wouldn't want to scare away potential customers worried their flights will get canceled.

If there is a sick-out but management and the union both deny it, how do they actually negotiate an agreement?

The denial by Southwest isn’t based on the same legal issues facing the union. The CEO has an incentive to deny it because he doesn’t want to risk losing the massive amounts of federal money Southwest receives.