What test will rule out candida esophagitis or laryngeal candidiasis in an otherwise health non HIV positive person? What specialists do these tests?

Inhaled steroids? Healthy people don't get candidal esophagitis. They might if they use inhaled steroids, or have rare immune deficiencies. Oral antibiotics might increase risk for oral candidal infection (thrush); shouldn't affect larynx. Ent docs can evaluate larynx, throat: less invasive than endoscopy. Candidal esophagitis typically is painful, and may cause swallowing issues.
An EGD, G.i. doctor. To get a 'good look' at the esophagus, a gastroenterologist (G.I.) doctor usually takes an 'endoscope' down the throat & esophagus for pictures and possible biopsy. You are sedated for the procedure (called an e.g.D.). The larynx is more accessible with a small scope through the nose, but this is done by an e.N.T. (ear-nose-throat). Hope this helps.