More than 9 million Americans deducted roughly $87 billion in medical expenses on their tax returns in 2015, according to the latest Internal Revenue Service data. If President Trump’s proposed tax changes become law, that write-off could be repealed.

One of those using the write-off is Paul Gregory, a 57-year-old resident of Plano, Texas, who was diagnosed with Amyotrophic Lateral Sclerosis five years ago. Mr. Gregory says that despite having insurance, he’s spending $130,000 per year out-of-pocket for round-the-clock care,...