My Chihuahua screams, shakes, won't eat, and acts like he's in pain.

Thank you for using Petco Pet Education Center, formerly Petcoach! Poor Rio, it sounds like he may be exhibiting this reaction due to pain. Some pets will display similar behavior if they are experiencing pain of the neck/back, abdomen, or mouth (dental). You are correct that these types of reactions can be due to severe anxiety, but this would most likely not affect his appetite and attitude when people are not interacting with him. I would recommend having him thoroughly evaluated by his veterinarian to see if the pain can be localized. Based on the physical examination findings, additional diagnostics may be recommended (complete blood count, serum chemistry, urinalysis), and treatments can be tailored based on findings.