Could the black and white wires in a three-wire cable both be used as hots?
This is no good

What's happening right now is that the bare grounding wire in the 10/3 feeder cable is being abused as a neutral by someone who clearly didn't know any better (as no-grounding-conductor feeders have not been Code for new installs for a while now).  You're going to need to re-terminate the cable (and probably use a box to splice some extra length in at one or both ends) and use it properly (black for one hot, red for the other hot, white for neutral, and bare for ground) in order to make this safe and Code compliant.

Also, if this is run using 10/3 cable and being fed by a 2-pole 60A breaker, then it's vastly over-breakered, and that breaker will need to be replaced with a 2-pole 30A unit while you're at it with this.

Now that you've mostly fixed it...

There's one last thing I'd like to point out, and that's the thing that looks to be a hole in the bottom left corner of the cutout box.  That hole should be able to take a 10-32 machine screw and pigtail, which you can then bundle in with the rest of the ground wires to ground the box.