Anxiety and depression are DEFINITELY conditions I have personally seen in horses before. My mare has been miserable since losing her gelding friend over a year ago. Whilst she copes with day to day actions, she is dull and yearns for other horsey socialising. We have two other horses on the property that "roam" and visit her from time to time, at least two times a week... and she goes back to normal, but when they leave her, she becomes depressed again. Financially I haven't been able to afford a second horse for her as company, but have recently looked into this for her.
As for anxiety, my gelding (that I sold) suffered from it. He was very sensitive to change. Even if you shifted the water tubs five inches from where they had just been, he'd refuse to drink for hours on end, snort fire at them, run up to them, then realise they still weren't where they had been, freak out, run away over and over until he finally accepted this. He did it with many other things too, except in the saddle - what a relief! He was an apprentice drivers horse (ex racing Standardbred) for 7yrs before I got him, so he had wound himself up and windsucked - though he never latch onto anything, simply twisted his lip to the side and sucked in air whenever he felt uncomfortable. He was healthy as apart from this issue (which had affected his teeth and thus he needed to live on longer grass and supplemented feeding). The anxiety episodes did slow down the longer I owned him, as he became accustomed to having one solid person in his life. But he still did suffer over the most random things... one thing I disliked was rugging him... he'd make SUCH a fuss on some days, on others not even a blink in recognition. Horses eh?!
I'm interested to see what comes of this interesting topic and to the OP, I hope you can get an answer soon... I'm sure it can be very frustrating and worrisome!