I have Hashi's and sometimes my hands are freezing after eating, sometimes not. If our temperature is low (cold hands) after eating a specific food, does that mean that it may not have been supportive and vice versa?

The thyroid is the furnace of the human body (i.e. - metabolism) and if you eat a food and feel unnaturally cold after eating it (obviously this does not apply to frozen foods, etc.) then it's an indicator that it's dragging down your metabolism (taking coal off the thyroid furnace) in order to process that food versus contributing to it (adding coal to keep the metabolic fire strong).