Hi Sookeh-
what's the top of your cage like.Is it the plastic base, metal top that you clip together?
When I got my boars I had one of those as an indoor cage and I put them out in it with a couple of heavy weights on it as an improvised run (they've got a rabbit run now). Have you got any metal tent pegs? That's what was supplied with our run to anchor ir down.
They won't burrow out- GPs are not burrowing animals- wrong sort of paws and far too lazy 
Can you catch them easily from the cage top (has it got a door or do you have to lift the whole thing?
Put a towel over one end to give them shade and a box to hide in.
YY to cats and foxes- that's their biggest danger (and magpies keep trying to traumatise my little black boar) but they'll run into their shelter if they're threatened. As long as you can't upturn the run.
And maybe give them a couple of hours to begin with? Check for buttercups on the grass and make sure they can't reach any bulb flowering plants.
Mine were out all day yesterday (GP2 had a bit of soft pooh, but that's him.)I give them a dry supper (no veg) so they were very judgy this morning.
Your not being over paranoid, honestly
they are a worry. Nothing worse than thinking "I shouldn't have done that "
But it gets easier, (like children, allegedly) .