It's been said in other threads, but bears have, allegedly, rarely, if ever been encountered above 9500 feet in the Uintas. At any rate, I can almost guarantee you'll never have to worry about bears along the Highline Trail. At least between Leidy and the Highline Trailhead of Mirror Lake Highway. If you're actually starting at Highway 191 then I can't vouch for that section leading up to Leidy. I too used to be worried about bears during my first trips into the High Uintas, but have since gotten much more lax about them.
Now days, I just bag all my food in a stuff sack lined with an odor proof barrier bag that I seal up, as I normally do, and then just hang it off a low lying branch in an attempt to keep it away from any possible rodents and other like critters since they're far more likely to smell out a bag of food at that elevation than a bear will. I've done many trips like this now and have not yet woken to find that my food bag had been tampered with. In fact, this is just how I'll be handling my food on an abbreviated version of the Highline Trail from Fox Lake to the west end with
@Parma next week.
So, I won't tell you not to take bear safety precautions, but based on my experience, the experience of others, and from what I've heard from rangers via 2nd hand sources (may have been
@Nick in another post), that 9500 foot line is a pretty safe guideline despite campground and trail head signs above which still say "This is Bear Country". To play it safer, you can even bump that guideline up to 10,000 feet which you'll still be above the whole way along the Highline Trail (assuming you're starting at Leidy or Chapeta). Given that for most black bears, their diet is 80-90% vegetarian and such primary food sources are far more abundant well below tree line and well below 9,500-10,000 feet, I just don't see any reason not to believe this guideline when it comes to bears in the Uintas.