You’ve correctly identified that it should have an axle with shoulders. The bearings are a slide fit on the axle and a light press fit in the hub shell. You tighten the nuts hard up against the bearings on these.
There’s a pretty good chance that most flip flop hubs use the same axles, because they’re mostly clones of each other and use the same bearings. But to be on the safe side you should get the same brand in case it’s not right. In your case though, it might be easiest to buy a whole identical hub, then you’ll get all the other bits you need ie set of bearings, lock nuts and proper track nuts. Bin the hub shell and put the cog and lock nut on eBay and it’ll cost about the same overall.