The reason gluten free diets are strongly suggested for those with hashimotos, is that the gluten molecule looks very similar in structure to some cells in the thyroid (strange I know). So when you have antibodies against your thyroid as with hashimotos, and you eat gluten, sometimes the body gets confused. The antibodies detect the gluten, and begin an attack, both towards the gluten molecule as well as thyroid tissue. This is why those who continue to eat gluten, can often have higher antibody readings.
Some people will pick up on these symptoms when they consume gluten, and remove it from their diet. However, because an immune attack can happen anywhere between hours to weeks, sometimes it can be difficult to determine why you are feeling awful some weeks and not others.
I choose to eat gluten free, and have since the moment I was diagnosed with hashimotos due to these reasons. My antibodies are low (originally >62, now 28). My mother, aunties and grandmother however had antibodies >300. With a gluten free diet, and supportive supplements, selenium, iron etc, all have reduced antibodies now <150.
I believe it's definietly worth a try! I know everyone is different, but with a condition such as hashimotos, that only tends to worsen over time- you have nothing to lose in trying