Got in a HU situation today where I had limped and both players checked on all streets and on the river I had nothing.
It got me thinking that he has two random cards and what probability it is for him to have hit. Can I call with only high card as a hand?

What are the chances that two random cards is anything more than air on the river?

Unless he does that with his entire range then he doesnt have a random hand here.

What was the board?

Yes but lets just assume that he does it with his entire range. How big of a bet does that allow me to call profitable?

A while ago but board was like king high with several low cards and my high card was a J.
I decided to fold. But I got very cruious about how often he really has a hand there.

Like simp said, we need to know the board, how much the bet is to call and how much is in the pot. Pokerstove can do the show you how your hand does vs atc on whatever the board was. (also if HU spot means just in that hand and not 2 left in tourney, you need to factor in icm)

But you can routinely call with a highcard heads up, esp vs villain who is betting atc.

J high seems like a good fold in that spot. But just realize that he's not doing this with his entire range so the exercise isnt that useful.